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    "# 11. Limits, Convergence, and Estimation\n",
    "\n",
    "## [Mathematical Statistical and Computational Foundations for Data Scientists](https://lamastex.github.io/scalable-data-science/360-in-525/2018/04/)\n",
    "\n",
    "©2018 Raazesh Sainudiin. [Attribution 4.0 International (CC BY 4.0)](https://creativecommons.org/licenses/by/4.0/)"
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    "# Inference and Estimation: The Big Picture\n",
    "\n",
    "- Limits\n",
    "  - Limits of Sequences of Real Numbers\n",
    "  - Limits of Functions\n",
    "  - Limit of a Sequence of Random Variables\n",
    "- Convergence in Distribution\n",
    "- Convergence in Probability\n",
    "- Some Basic Limit Laws in Statistics\n",
    "- Weak Law of Large Numbers\n",
    "- Central Limit Theorem\n",
    "  \n",
    "\n",
    "### Inference and Estimation: The Big Picture\n",
    "\n",
    "The Markov Chains we discussed earlier fit into our Big Picture, which is about inference and estimation and especially inference and estimation problems where computational techniques are helpful. \n",
    "\n",
    "<table border=\"1\" cellspacing=\"2\" cellpadding=\"2\" align=\"center\">\n",
    "<tbody>\n",
    "<tr>\n",
    "<td style=\"background-color: #ccccff;\" align=\"center\">&nbsp;</td>\n",
    "<td style=\"background-color: #ccccff;\" align=\"center\"><strong>Point estimation</strong></td>\n",
    "<td style=\"background-color: #ccccff;\" align=\"center\"><strong>Set estimation</strong></td>\n",
    "</tr>\n",
    "<tr>\n",
    "<td style=\"background-color: #ccccff;\">\n",
    "<p><strong>Parametric</strong></p>\n",
    "<p>&nbsp;</p>\n",
    "</td>\n",
    "<td style=\"background-color: #ccccff;\" align=\"center\">\n",
    "<p>MLE of finitely many parameters<br /><span style=\"color: #3366ff;\"><em>done</em></span></p>\n",
    "</td>\n",
    "<td style=\"background-color: #ccccff;\" align=\"center\">\n",
    "<p>Confidence intervals,<br /> via the central limit theorem</p>\n",
    "</td>\n",
    "</tr>\n",
    "<tr>\n",
    "<td style=\"background-color: #ccccff;\">\n",
    "<p><strong>Non-parametric</strong><br /> (infinitely many parameters)</p>\n",
    "</td>\n",
    "<td style=\"background-color: #ccccff;\" align=\"center\"><em><span style=\"color: #3366ff;\">coming up ... </span></em></td>\n",
    "<td style=\"background-color: #ccccff;\" align=\"center\">&nbsp;<span style=\"color: #3366ff;\"><em>coming up ...</em></span></td>\n",
    "</tr>\n",
    "<tr>\n",
    "<td style=\"background-color: #ccccff;\">\n",
    "<p><strong>One/Many-dimensional Integrals</strong><br /> (finite-dimensional)</p>\n",
    "</td>\n",
    "<td style=\"background-color: #ccccff;\" align=\"center\"><em><span style=\"color: #3366ff;\">coming up ... </span></em></td>\n",
    "<td style=\"background-color: #ccccff;\" align=\"center\">&nbsp;<span style=\"color: #3366ff;\"><em>coming up ...</em></span></td>\n",
    "</tr>\n",
    "</tbody>\n",
    "</table>\n",
    "\n",
    "But before we move on we have to discuss what makes it all work: the idea of limits - where do you get to if you just keep going?\n",
    "\n",
    "## Limits\n",
    "\n",
    "Last week we described a Markov Chain, informally, as a system which \"jumps\" among several states, with the next state depending (probabilistically) only on the current state.  Since the system changes randomly, it is generally impossible to predict the exact state of the system in the future.  However, the statistical and probailistic properties of the system's future can be predicted.  In many applications it is these statistical properties that are important.   We saw how we could find a steady state vector:\n",
    "\n",
    "$$\\mathbf{s} = \\lim_{n \\to \\infty} \\mathbf{p}^{(n)}$$\n",
    "\n",
    "(And we noted that $\\mathbf{p}^{(n)}$ only converges to a strictly positive vector if $\\mathbf{P}$ is a regular transition matrix.)\n",
    "\n",
    "The week before, we talked about the likelihood function and maximum likelihood estimators for making point estimates of model parameters.  For example for the $Bernoulli(\\theta^*)$ RV (a $Bernoulli$ RV with true but possibly unknown parameter $\\theta^*$, we found that the likelihood function was $L_n(\\theta) = \\theta^{t_n}(1-\\theta)^{(n-t_n)}$ where $t_n = \\displaystyle\\sum_{i=1}^n x_i$.  We also found the maxmimum likelihood estimator (MLE) for the $Bernoulli$ model, $\\widehat{\\theta}_n = \\frac{1}{n}\\displaystyle\\sum_{i=1}^n x_i$. \n",
    "\n",
    "We demonstrated these ideas using samples simulated from a $Bernoulli$ process with a secret $\\theta^*$.  We had an interactive plot of the likelihood function where we could increase $n$, the number of simulated samples or the amount of data we had to base our estimate on, and see the effect on the shape of the likelihood function.  The animation belows shows the changing likelihood function for the Bernoulli process with unknown $\\theta^*$ as $n$  (the amount of data) increases.\n",
    "\n",
    "\n",
    "<table style=\"width:100%\">\n",
    "  <tr>\n",
    "    <th>Likelihood function for Bernoulli process, as $n$ goes from 1 to 1000 in a continuous loop.</th>\n",
    "  </tr>\n",
    "<tr>\n",
    "    <th><img src=\"images/bernoulliLikelihoodAnim.gif\" width=300></th>\n",
    "  </tr>\n",
    "</table>\n",
    "\n",
    "For large $n$, you can probably make your own guess about the true value of $\\theta^*$ even without knowing $t_n$.  As the animation progresses, we can see the likelihood function 'homing in' on $\\theta = 0.3$. \n",
    "\n",
    "We can see this in another way, by just looking at the sample mean as $n$ increases.  An easy way to do this is with running means:  generate a very large sample and then calculate the mean first over just the first observation in the sample, then the first two, first three, etc etc (running means were discussed in an earlier worksheet if you want to go back and review them in detail in your own time).  Here we just define a function so that we can easily generate sequences of running means for our $Bernoulli$ process with the unknown $\\theta^*$."
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    "#### Preparation: Let's just evaluate the next cel and focus on concepts.\n",
    "\n",
    "You can see what they are as you need to."
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    "def likelihoodBernoulli(theta, n, tStatistic):\n",
    "    '''Bernoulli likelihood function.\n",
    "    theta in [0,1] is the theta to evaluate the likelihood at.\n",
    "    n is the number of observations.\n",
    "    tStatistic is the sum of the n Bernoulli observations.\n",
    "    return a value for the likelihood of theta given the n observations and tStatistic.'''\n",
    "    retValue = 0 # default return value\n",
    "    if (theta >= 0 and theta <= 1): # check on theta\n",
    "        mpfrTheta = RR(theta) # make sure we use a Sage mpfr \n",
    "        retValue = (mpfrTheta^tStatistic)*(1-mpfrTheta)^(n-tStatistic)\n",
    "    return retValue\n",
    "    \n",
    "def bernoulliFInverse(u, theta):\n",
    "    '''A function to evaluate the inverse CDF of a bernoulli.\n",
    "    \n",
    "    Param u is the value to evaluate the inverse CDF at.\n",
    "    Param theta is the distribution parameters.\n",
    "    Returns inverse CDF under theta evaluated at u'''\n",
    "    \n",
    "    return floor(u + theta)\n",
    "    \n",
    "def bernoulliSample(n, theta, simSeed=None):\n",
    "    '''A function to simulate samples from a bernoulli distribution.\n",
    "    \n",
    "    Param n is the number of samples to simulate.\n",
    "    Param theta is the bernoulli distribution parameter.\n",
    "    Param simSeed is a seed for the random number generator, defaulting to 30.\n",
    "    Returns a simulated Bernoulli sample as a list.'''\n",
    "    \n",
    "    set_random_seed(simSeed)\n",
    "    us = [random() for i in range(n)]\n",
    "    set_random_seed(None)\n",
    "    return [bernoulliFInverse(u, theta) for u in us] # use bernoulliFInverse in a list comprehension\n",
    "    \n",
    "def bernoulliSampleSecretTheta(n, theta=0.30, simSeed=30):\n",
    "    '''A function to simulate samples from a bernoulli distribution.\n",
    "    \n",
    "    Param n is the number of samples to simulate.\n",
    "    Param theta is the bernoulli distribution parameter.\n",
    "    Param simSeed is a seed for the random number generator, defaulting to 30.\n",
    "    Returns a simulated Bernoulli sample as a list.'''\n",
    "    \n",
    "    set_random_seed(simSeed)\n",
    "    us = [random() for i in range(n)]\n",
    "    set_random_seed(None)\n",
    "    return [bernoulliFInverse(u, theta) for u in us] # use bernoulliFInverse in a list comprehension\n",
    "\n",
    "def bernoulliRunningMeans(n, myTheta, mySeed = None):\n",
    "    '''Function to give a list of n running means from bernoulli with specified theta.\n",
    "    \n",
    "    Param n is the number of running means to generate.\n",
    "    Param myTheta is the theta for the Bernoulli distribution\n",
    "    Param mySeed is a value for the seed of the random number generator, defaulting to None.'''\n",
    "     \n",
    "    sample = bernoulliSample(n, theta=myTheta, simSeed = mySeed)\n",
    "    from pylab import cumsum # we can import in the middle of code\n",
    "    csSample = list(cumsum(sample))\n",
    "    samplesizes = range(1, n+1,1)\n",
    "    return [RR(csSample[i])/samplesizes[i] for i in range(n)]\n",
    "    \n",
    "#return a plot object for BernoulliLikelihood using the secret theta bernoulli generator\n",
    "def plotBernoulliLikelihoodSecretTheta(n):\n",
    "    '''Return a plot object for BernoulliLikelihood using the secret theta bernoulli generator.\n",
    "    \n",
    "    Param n is the number of simulated samples to generate and do likelihood plot for.'''\n",
    "    \n",
    "    thisBSample = bernoulliSampleSecretTheta(n) # make sample\n",
    "    tn = sum(thisBSample) # summary statistic\n",
    "    from pylab import arange\n",
    "    ths = arange(0,1,0.01) # get some values to plot against\n",
    "    liks = [likelihoodBernoulli(t,n,tn) for t in ths] # use the likelihood function to generate likelihoods\n",
    "    redshade = 1*n/1000 # fancy colours\n",
    "    blueshade = 1 - redshade\n",
    "    return line(zip(ths, liks), rgbcolor = (redshade, 0, blueshade))\n",
    "    \n",
    "def cauchyFInverse(u):\n",
    "    '''A function to evaluate the inverse CDF of a standard Cauchy distribution.\n",
    "    \n",
    "    Param u is the value to evaluate the inverse CDF at.'''\n",
    "    \n",
    "    return RR(tan(pi*(u-0.5)))\n",
    "    \n",
    "def cauchySample(n):\n",
    "    '''A function to simulate samples from a standard Cauchy distribution.\n",
    "    \n",
    "    Param n is the number of samples to simulate.'''\n",
    "    \n",
    "    us = [random() for i in range(n)]\n",
    "    return [cauchyFInverse(u) for u in us]\n",
    "\n",
    "def cauchyRunningMeans(n):\n",
    "    '''Function to give a list of n running means from standardCauchy.\n",
    "    \n",
    "    Param n is the number of running means to generate.'''\n",
    "    \n",
    "    sample = cauchySample(n)\n",
    "    from pylab import cumsum\n",
    "    csSample = list(cumsum(sample))\n",
    "    samplesizes = range(1, n+1,1)\n",
    "    return [RR(csSample[i])/samplesizes[i] for i in range(n)]\n",
    "\n",
    "def twoRunningMeansPlot(nToPlot, iters):\n",
    "    '''Function to return a graphics array containing plots of running means for Bernoulli and Standard Cauchy.\n",
    "    \n",
    "    Param nToPlot is the number of running means to simulate for each iteration.\n",
    "    Param iters is the number of iterations or sequences of running means or lines on each plot to draw.\n",
    "    Returns a graphics array object containing both plots with titles.'''\n",
    "    xvalues = range(1, nToPlot+1,1)\n",
    "    for i in range(iters):\n",
    "        shade = 0.5*(iters - 1 - i)/iters # to get different colours for the lines\n",
    "        bRunningMeans = bernoulliSecretThetaRunningMeans(nToPlot)\n",
    "        cRunningMeans = cauchyRunningMeans(nToPlot)\n",
    "        bPts = zip(xvalues, bRunningMeans)\n",
    "        cPts = zip(xvalues, cRunningMeans)\n",
    "        if (i < 1):\n",
    "            p1 = line(bPts, rgbcolor = (shade, 0, 1))\n",
    "            p2 = line(cPts, rgbcolor = (1-shade, 0, shade))\n",
    "            cauchyTitleMax = max(cRunningMeans) # for placement of cauchy title\n",
    "        else:\n",
    "            p1 += line(bPts, rgbcolor = (shade, 0, 1))\n",
    "            p2 += line(cPts, rgbcolor = (1-shade, 0, shade))\n",
    "            if max(cRunningMeans) > cauchyTitleMax: cauchyTitleMax = max(cRunningMeans)\n",
    "    titleText1 = \"Bernoulli running means\" # make title text\n",
    "    t1 = text(titleText1, (nToGenerate/2,1), rgbcolor='blue',fontsize=10) \n",
    "    titleText2 = \"Standard Cauchy running means\" # make title text\n",
    "    t2 = text(titleText2, (nToGenerate/2,ceil(cauchyTitleMax)+1), rgbcolor='red',fontsize=10)\n",
    "    return graphics_array((p1+t1,p2+t2))\n",
    "\n",
    "def pmfPointMassPlot(theta):\n",
    "    '''Returns a pmf plot for a point mass function with parameter theta.'''\n",
    "    \n",
    "    ptsize = 10\n",
    "    linethick = 2\n",
    "    fudgefactor = 0.07 # to fudge the bottom line drawing\n",
    "    pmf = points((theta,1), rgbcolor=\"blue\", pointsize=ptsize)\n",
    "    pmf += line([(theta,0),(theta,1)], rgbcolor=\"blue\", linestyle=':')\n",
    "    pmf += points((theta,0), rgbcolor = \"white\", faceted = true, pointsize=ptsize)\n",
    "    pmf += line([(min(theta-2,-2),0),(theta-0.05,0)], rgbcolor=\"blue\",thickness=linethick)\n",
    "    pmf += line([(theta+.05,0),(theta+2,0)], rgbcolor=\"blue\",thickness=linethick)\n",
    "    pmf+= text(\"Point mass f\", (theta,1.1), rgbcolor='blue',fontsize=10)\n",
    "    pmf.axes_color('grey') \n",
    "    return pmf\n",
    "    \n",
    "def cdfPointMassPlot(theta):\n",
    "    '''Returns a cdf plot for a point mass function with parameter theta.'''\n",
    "    \n",
    "    ptsize = 10\n",
    "    linethick = 2\n",
    "    fudgefactor = 0.07 # to fudge the bottom line drawing\n",
    "    cdf = line([(min(theta-2,-2),0),(theta-0.05,0)], rgbcolor=\"blue\",thickness=linethick) # padding\n",
    "    cdf += points((theta,1), rgbcolor=\"blue\", pointsize=ptsize)\n",
    "    cdf += line([(theta,0),(theta,1)], rgbcolor=\"blue\", linestyle=':')\n",
    "    cdf += line([(theta,1),(theta+2,1)], rgbcolor=\"blue\", thickness=linethick) # padding\n",
    "    cdf += points((theta,0), rgbcolor = \"white\", faceted = true, pointsize=ptsize)\n",
    "    cdf+= text(\"Point mass F\", (theta,1.1), rgbcolor='blue',fontsize=10)\n",
    "    cdf.axes_color('grey') \n",
    "    return cdf\n",
    "    \n",
    "def uniformFInverse(u, theta1, theta2):\n",
    "    '''A function to evaluate the inverse CDF of a uniform(theta1, theta2) distribution.\n",
    "    \n",
    "    u, u should be 0 <= u <= 1, is the value to evaluate the inverse CDF at.\n",
    "    theta1, theta2, theta2 > theta1, are the uniform distribution parameters.'''\n",
    "    \n",
    "    return theta1 + (theta2 - theta1)*u\n",
    "\n",
    "def uniformSample(n, theta1, theta2):\n",
    "    '''A function to simulate samples from a uniform distribution.\n",
    "    \n",
    "    n > 0 is the number of samples to simulate.\n",
    "    theta1, theta2 (theta2 > theta1) are the uniform distribution parameters.'''\n",
    "    \n",
    "    us = [random() for i in range(n)]\n",
    "    \n",
    "    return [uniformFInverse(u, theta1, theta2) for u in us]\n",
    "\n",
    "def exponentialFInverse(u, lam):\n",
    "    '''A function to evaluate the inverse CDF of a exponential distribution.\n",
    "    \n",
    "    u is the value to evaluate the inverse CDF at.\n",
    "    lam is the exponential distribution parameter.'''\n",
    "    \n",
    "    # log without a base is the natural logarithm\n",
    "    return (-1.0/lam)*log(1 - u)\n",
    "    \n",
    "def exponentialSample(n, lam):\n",
    "    '''A function to simulate samples from an exponential distribution.\n",
    "    \n",
    "    n is the number of samples to simulate.\n",
    "    lam is the exponential distribution parameter.'''\n",
    "    \n",
    "    us = [random() for i in range(n)]\n",
    "    \n",
    "    return [exponentialFInverse(u, lam) for u in us]"
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    "To get back to our running means of Bernoullin RVs:"
   ]
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    "def bernoulliSecretThetaRunningMeans(n, mySeed = None):\n",
    "    '''Function to give a list of n running means from Bernoulli with unknown theta.\n",
    "    \n",
    "    Param n is the number of running means to generate.\n",
    "    Param mySeed is a value for the seed of the random number generator, defaulting to None\n",
    "    Note: the unknown theta parameter for the Bernoulli process is defined in bernoulliSampleSecretTheta\n",
    "    Return a list of n running means.'''\n",
    "    \n",
    "    sample = bernoulliSampleSecretTheta(n, simSeed = mySeed)\n",
    "    from pylab import cumsum # we can import in the middle of code\n",
    "    csSample = list(cumsum(sample))\n",
    "    samplesizes = range(1, n+1,1)\n",
    "    return [RR(csSample[i])/samplesizes[i] for i in range(n)]"
   ]
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    "Now we can use this function to look at say 5 different sequences of running means (they will be different, because for each iteration, we will simulate a different sample of $Bernoulli$ observations). "
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FRKQxKEiLiIg0UwrSbuDKpDXjmIiInA7NOOYGPaKz2ZChGcdEROT0KJMWERFpphSk3UAD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\nhBDCUzSkCSGEEJ76DyX9/sqWXmcHAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "Graphics object consisting of 5 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nToGenerate = 1500\n",
    "iterations = 5\n",
    "xvalues = range(1, nToGenerate+1,1)\n",
    "for i in range(iterations):\n",
    "    redshade = 0.5*(iterations - 1 - i)/iterations # to get different colours for the lines\n",
    "    bRunningMeans = bernoulliSecretThetaRunningMeans(nToGenerate)\n",
    "    pts = zip(xvalues,bRunningMeans)\n",
    "    if (i == 0):\n",
    "        p = line(pts, rgbcolor = (redshade,0,1))\n",
    "    else:\n",
    "        p += line(pts, rgbcolor = (redshade,0,1))\n",
    "show(p, figsize=[5,3], axes_labels=['n','sample mean'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "What we notice is how the different lines **converge** on a sample mean of close to 0.3. \n",
    "\n",
    "Is life always this easy?  Unfortunately no.  In the plot below we show the well-behaved running means for the $Bernoulli$ and beside them the running means for simulated  standard $Cauchy$ random variables.  They are all over the place, and each time you re-evaluate the cell you'll get different all-over-the-place behaviour. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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NGDBAaWmVz7T49ttv64EHHtBjjz2mtWvXatq0aZo1a5YeeuihIw4eAAAA8GWW\nZanZO5MV8eZ/ZBcVK/3cq5R5z+NOhwUckaJvVmln885KH3yjPMxLcFDVTrYnTZqkkSNHatiwYerU\nqZMmT56s0NBQTZs2rdLtv/nmG5155pm65ppr1Lp1a/Xv319DhgzRd999d8TBAwAAAP4g5LorFL1x\nmayWccp7dooyxzzodEhAjRUu/EL2nkwVfrhAqcecot0XXafin9c6HZbPqVayXVxcrFWrVqlfv377\n1rlcLvXv31/ffPNNpfv06dNHq1at2tfVfNOmTZo7d64uueSSIwgbAAD4isTERCUkJCgpKcnpUACf\n5m7dUtFbvlXAsW2U99Ibyrz9EadDAo5I0zeel3VMCxXN/1xpXftph6ul0npdosKl3zodmk9wV2fj\ntLQ0lZSUKDY2tsL62NhYrVu3rtJ9hgwZorS0NJ155pmybVslJSUaNWqU7rvvvppHDQAAfEZycrLC\nw8OdDgPwC5bbrahfPlPq8Wcr74Vpynthmhqdcaoi3psqd1yM0+EB1eI+6QTF/rFCnvWblD3+GRW8\nP0/FK35Q+lmXy93leDV7/1W5OxzrdJiOqZXZyG3blusgz+xasmSJnnzySU2ePFmrV6/W+++/r48/\n/lgTJ048bLkdOnRQXFycevbsqYSEBO6aAwAAwO9ZwcGK3rxcgeefLUkq/nqFUludopwXXnM4MqBm\n3B2OVbOklxVfuEUxKWvUqHcPeX5ep9SOZymlZQ+lD75RRSvXOB1mnatWy3ZUVJQCAgKUkpJSYf2u\nXbsOaO0uNX78eA0bNkzDhw+XJJ144onKycnRyJEj9fDDDx/yeOvXr+dOOQAAAOody7IUuTBJnnUb\nlfP0Syr438fKvn28ipYsV8S7U2RZPKEX/ikgJkpRy+eo+Jd1yn74aRXOX6LCDxeo8MMFsmKjFHLT\nUIVNGCvLXa1U1C9V639xo0aN1LNnTy1evHjfOtu2tXjxYvXp06fSffLy8g74srAsS7Zty7btGoQM\nAAAA1A/u49srYtpzikn9Se6TOqvw/bnKe/Zlp8MCjlijE49X8w+mKT5/kyLeflFBgy+UnZ2r3In/\n1p6BNzgdXp2o9i2zsWPHaurUqZoxY4bWrl2rUaNGKS8vTzfccIMkadiwYXrwwbLZFQcOHKiXX35Z\ns2bN0pYtW/Tpp59q/PjxGjRo0EG7ngMAAAANiRUYqMhV86RGbmXf96R2HddH3rR0p8MCakXIkMvU\n/P3XFJdsUeTYAAAgAElEQVS7Qa5mTVU0/3PtDOugPdeOkbeoyOnwjppqt91fffXVSktL0/jx45WS\nkqKTTz5ZCxYsUHR0tCRp+/btcpfrEvDII4/Isiw98sgj+vPPPxUdHa2EhISDj9km/wYAAEADZLnd\nil73pdJOvUQlG7cqpUUPhb/0hBqPGOp0aECtidm+UukXXqfir79TQdKHKnj3YwWe1UtNnn5Qgad2\ndzq8WuWyfawvd162dEV4luarqTIzMxmzDQCAj8rKylLTptTXcFbRDz9rd/cBCnt0nJpMGOt0OLWm\nYO5iZVw9SnZuntSokZovSlbQ2ac5HRYgScp+7F/KefQ5Ra5eoMCTu9SoDK/Xq7znJiv3+Vfl/cvM\nCWbFRCo4cZDcnToo5PorZYWG1mbYdY6ZFwAAAOC/vD7VblRrgi/up5i0n9SoV3epuFjpfa9Q3hvv\nOB0WYNjeIy7CsiyF3TNasX9+r+htKxV87WDZOXnK+880ZY1+QClhHZXa5VzlvPCavB5PLQRd90i2\nAQAAAB9kBQcr6tuPFbl8jlyNQ5V5w11KO/ViebOynA4NMGpp1nx3q3g1e+u/isvdYCZTu7S/Gp3S\nTZ61G5V9+3ilBLdT6ikXKW/Gu/J6jzzRrysk2wAAAIAPC+zdQzE7V8vds6uKV65RStMTVPT9T06H\nBRwVIUMuU/M5byjqu08UW7BJ4S89JXfnjvJ8/5Myr79TKUHtlNrtfGVPfF7ejEynwz0kkm0AAADA\nx1lhYYpeOV+hd4+UJO0+5SJl3jXB4aiAo8tyu9X4lmGK/nGxYvM2qMlTDyigfRt5flmnnEeeUUqz\nztoR2FY7Gx+nzLGP+twM/iTbAAAAgJ9o+sx4Rf/+laxW8cp7/lXt6nimvOl7nA4LOOqs4GCF3X+r\nYtZ+qdiiLYr46HUFXdJfrtBg2Xn5ypv0ilKiu2pn+PHyrN/kdLiSSLYBAAAAv+LucKyit3yr4Csv\nVcn6zUqJ7KLMex53OiygzliWpZCEC9T84zcUl7FW8fafajb3TQUc11Z2do5SO56l1JP6KfvxSfLm\n5DgXp2NHBgAAAFAjlmWp2btT1Gzh21JQoPKenaJd7fvIs36Tir79Xrn/fd2vJpICjlTwRecpZv3X\nivzuY7m7niDPz+uUM+FZpTQ5XruO66Psx/5V593MSbYBAAAAPxV8fl/F5m1UyIghKtm0VanHn63d\npw1U1m0Pa1fMScr/YJ7TIQJ1KvDU7or+cZFiCzYpYtbLanR2b5Vs+0s5jz6nlOiu2uE+Rml9L1f+\nB/OO+g0pkm0AAADAj1mWpYhXnlXkmk9ltW65b72dma2My0doR6M2ynnmJQcjBOqeFRiokKsTFPXF\n+4rN36SI96bK3eV4WS3iVPzlt8q4fIRSQo5VavcLlDX+GXn+2ln7MdR6iQAAoEFJTExUQkKCkpKS\nnA4FaNACT+qs2C3fKnDAOQq9c4Ri9/yi4CsukTweZd/7hHa4Wirjhjsr7FPw6RdKPaEvLeCo1yzL\nUsgVlyj6p88U+8cKxWb+prAJdymg7THy/LRWuX9/Xqkte2pneEel9b281oZhuGsh9lrlcjkdAQAA\nqI7k5GSFh4c7HQaAvSLnv7XvdbP3psrr9Sp73GPKe/5V5b/xrjy/rlfEh6/J3SJO+dPflWftBmVc\nPkLZHdop4q0XFHhqdwejhz+wvbbTIRwRKzxcTR69W00evVter1fFn3+tvNeSVfTlchV/9Z2Kl61U\n41uHH/lxaiFWAAAAwFm02ByUZVlqOukxxWb+psB+Z6p4xQ9KPeZUZY5+QPKWSJKCr7hEJRu2aHev\nS5V2+kB5tm5zOGr4A5fl/+mkZVkK6neWmr39omK3r1Js0RZFLv2wdsqulVIAAAAA+DQrPFyRi2Yp\nctlsWfExynt5hgpmzZZkWsCj/1ihRmf2UvHy75Xa7nSlXzZc3qwsh6MG6pbldiuwd4/aKatWSgEA\nAADgFwJP76nY7avU5LnxUmCQtLd10t0qXlFffaDIVfMVcHx7FX600DzDe8yD8no8DkcN+B+SbQAA\nAKABChs7UrEZvyp68/IK6wN7dFXMb18oYs4bsqIilffSG0oJaa+cf77oUKSAfyLZBgAAABooKzhY\n7nKPCysv5NL+it2xWuEvPSW5A5R935NK7XyOPJu31nGUgH/y6WSbR4kAAAAAzmp8yzBFb1ymwAHn\nyPPbeqW2P0O7+18jzx9/Oh0a4NN87tFf5fEoEQAAAMB57hZxipz/lopWrlHm/92tosVLldq2twIv\nPFfN3v6vrIimTocI+ByfbtkGAAAA4DsCT+mm6DWfKnLVfLm7HK+ieZ8pJbKL9gy5hZnLgf2QbAMA\nAAColsAeXRX942I1W/i2rGNaqCB5tlKad9GOgGOUcfM98hYUOB0i4DiSbQAAAAA1Enx+X8Vu+VYR\nc96QKzxM8nqV/8rbSmlyvDKG3yVvXp7TIeJosG2nI/ALPpdsu1xORwAAAACgOkIu7a+49F8VuWq+\nIpJfkhUbpfzp7yglvJP2DL2VLub1FcnbIflcsg0AAADAPwX26KqQawYpdvsqRbz/qqwWsSp4+wOl\nND9Re64eJW9GptMhAnWGZBsAAAD+y+t1OgIcRMjgixT7xwpFzHlDAa1bquDdOUqJ7KLdF12n4p/X\nOh0ecNSRbAMAAAA4akIu7a+YTcvVbOHbCji2tYrmf660rv20M7qrMu95XN69N0x2RpygnREnyFtU\n5HDEQO0g2QYAAABw1AWf31cx679W9IavFXRxP9lp6cp7dopSQttrz1UjZWdmyc7MUkrTTsq87WF5\n0/c4HTJwREi2AQDAEUlMTFRCQoKSkpKcDgWAH3C3b6vmn8xQbP5GNXnqAVlRzVXw3seSJKtNK7lC\ngpX339eVEtVVnr92OhwtUHMk2wAA4IgkJydr9uzZGjJkiNOhAPAjVnCwwu6/VbHbV6n50g8VdNkA\nNf/4DcWl/6rG94+RbFuprU5R2hmDVPjZUqfDBaqNZBsAAACAo4LOOFXNP5imRl06SZLCn3pQEe+/\nKneXTipetlLp/a7RzohOyhgxjmd3w2+QbAMAAADwOSGDL1L0j4sUk7JGobcMk9xu5b+WrF3HnKqC\nBUucDg84LJJtAAAAAD4rICZKTV96SnFpPyv4moGy0zO058Kh2hneUZl3TWD2cid4bacj8As+mWwz\n7yAAAACA/TVLnqzY3T8r9NbhUkCA8p5/VSmNj1P64BuZTM0Jlk+mkz7DJ89OodMBAAAAAPBJVvNm\navrCRMXt+U3h056T1aqFCj9coNSWPZV6Ql/lzXh337O7ASf5XLLtcjkdAQAAAAB/0Hh4omI3L1fk\n8jlqdHZvedZvUub1dyolqJ3Szhik/NkLnQ4RDZjPJdsAAAAAUB2BvXso6ov3FZu3UWGPjlPAsa1V\n/M1KZQwarp1hHbQncZRyX5quHcHHaveFQ5nRHHWCZBsAAABAvWAFBqrJhLGKWfeVYjN+U+g9t8jV\npLEKZs1R1piHpMJCFS1YopRmnZX1yD/lLShwOmTUYyTbAAAAAOodKzxcTf/5sGJ3/KCI96bK3bmj\nrGNaqMnTD0ouS7kT/609F//N6TBRj5FsAwAAwP9ZTPyDgwu54hJF//K5Yv9YobB7xyi+YJOs+FgV\nfb5Muy8YouJf1jkdIuohn062ExMTlZCQoKSkJKdDAQAAh+D54095du5yOgwAqLLmi2cpoEM7FX36\npdK6nKeU1qcq864JKtmV5nRoqCfcTgdwKMnJyQoPD3c6DAAAcBipbXpJLpfivdudDgUAqqTRCR0U\n8/tSM4P5HRNUtGSZ8p5/VXnPvyorJkqNTj1ZIcOvVtDgi2TxPGnUAH81AACgdti20xEAQLW5Oxyr\nyLlvKj5vo5p//q6CLu4nu7hYhZ8sUsaVNyslrIPSB16vgnmf8fzuUnzfVwnJNgAAAABICjqnj5p/\nMkNx6b8qNvM3NX74DllhjVX48SLtufhvSglqp4xb7ifp3stFi/8hcXYAAAAAYD9WeLjC/36vYnf9\nqNjUn9T4kTsld4DyJ7+plJBjlXHL/fJs3+F0mPBhJNsAAOCIJCYm6nrt1gfKczoUADgqrKjmCn/8\nHsXuWK3G946WKyhQ+ZPfVOoxpygltpsybrpbhZ8tpcUbFfhess1TGwAA8CvJycl6Q5EarFCnQwGA\no8qKaKrwpx9SXNbvilw2W0GXDZCdX6D8V5OU3u8apQS2VVqfBOW/M5vEGz6YbAMAAACAjws8vaea\nfzBNcVnrFLliroIu7qeA9m1UvPx7ZVxzi1JC2mv3+Ykq/m2906HCISTbAAAA8F82rYdwXuAp3dT8\nkxmKWfeVYrPWKuzRcQpoGauiRV8prfM52hl5otIH3aDCz5Y6HSrqEMk2AAAAANQSKyxMTSaMVcym\n5YpcvUCB550hSSqc/anS+12jHcHtlNYnQbmvJcnr8TgcLY4mkm0AAAAAOAoCT+6iyMXvKG73L4re\nO7laQKt4FS//Xlkj7lZKUDvtOqGvsh99Vp6NW+QtKnI6ZNQikm0AAAAAOMrccTEKf/ohxWxYpti8\nDQp/YaLc3buoZOMW5Tw2SanHnaGUoHZK7TlAuf99ncS7HiDZBgAAAIA6ZAUHq/GtwxW9cp5iCzYr\nYs4batTnFFmt4uX54Vdl3fawUoKPVUrrU5Ux/C4VzF0sb54PPV6RuRKqxO10AAAAAADQUFmWpZBL\n+yvk0v6SJK/Ho/zps5Q//V15fvhF+dPfUf70dyRJoaOvl/vEjmrUq7sCT+nmZNiGxXObD4VkGwAA\nAAB8hOV2q/GIoWo8YqgkybN1m/Kmvq3cJ/+jvJfeKNvQ7ZaraRPZ6RkK6NBOoTdcreDh18gdF+NQ\n5Ngf3cgBAAAAwEe52xyj8CfuU2zxVkW8N1XBVw1U6G03yt25o+zsHMm2VfL7JmU/+A+lxnfXjpBj\nldr5HGXcdLcKv1zudPgNGi3bAAAAAODjLLdbIVdcopArLjngPa/Ho8JPFqnwf3NVtPx7eTZtlee3\n9cp/NUlyuxV0yXkKGX6NggZeIMuivbWucKYBAIAk6bHHHpNlWRX+de7c2emwAACHYbndChl0oSJm\n/Ecxvy9VfMFmRW9ZrsDzz5arcYgKP1qojMv+TymBbbSr45nKuPkeedZtdDrses/nWrZdjLEHAMAx\nXbp00eLFi2XbtiTJ7T74pULBwiWSpMIly+oiNABANbjbHKPIhUmSJG/6HuW+/KYK/veJPL+uV/4r\nbyv/lbfl7tpJoWNuUMjfrpACA2Ud4jsf1UfLNgAA2Mftdis6OloxMTGKiYlR8+bND7ptfvJH5ud7\nH9dVeACAGrCaN1OTh25X9PcLFF+wSWGP3S25XPL8tFZZo+5XSuMOSmnURrtO6KvMex6XNyPT6ZDr\nBZJtAACwz/r169WyZUu1b99e1113nbZt23bwjV17LyPKPW7V6+XZqwDg65qMv0uxnj8Uu+dXhb/6\nrAL7nyUFBKhk01blPTtFKc06a2dEJ6WdfblyX0uS1+NxOmS/RLINAAAkSaeddpqmT5+uBQsWaPLk\nydq8ebPOPvts5ebmVr5DZWO/ioqObpAAgFphWZasiKZq/H9DFPlpsuI9fyi+cIuazX1TQZf2l6tx\nqIqXfqesEXcrpVEb7QzroLS+lyv3pemyc/KcDt8v+HSn/MTERLndbg0ZMkRDhgxxOhwAAOq1AQMG\n7HvdpUsX9erVS23atNE777yj4cOHH3S//1vysVzaLUkKuvxyuai7AcBvBV90noIvOk+S5C0oUN6L\n01X01bcq/v5nFX/5rYq//NbhCP2HTyfbycnJCg8PdzoMAAAapKZNm6pjx47asGFDpe+79rZsv3b2\nRWqUNFuSFDvzLVnNm9VZjACAo8cKDlbYuFHSuFGSTPJd+NECFXw4X96MbFnt2zoboI/z6WQbAAA4\nJycnRxs3btSwYcMq36CSbuS2p+QoRwUAcIoVHKyQawYp5JpBTofiFxizDQAAJEn33HOPvvzyS23d\nulXLli3T4MGD9w3nOqS9jwmTJLug8ChHCRyEi8taAL6Flm0AACBJ2r59u6699lrt3r1b0dHROvPM\nM7V8+XJFRkZWvkMl86PZ2TlHN0gAAPwEyTYAAJAkJSUl1WxHu9zLnIPMXA4AQANDfxsAAFAzpWO2\ny3Uj92ZlOxQMAAC+xeeS7coe2QkAAHxQZROkZdGNHAAAyQeTbQAA4F/s8hOk0Y0cAABJJNsAAKCm\nSmd/tu19rdx2bp6DAaFB8tqH3wYAHECyDQAAaqb8VYS1N9mmZRsAAEkk2wAA4EjZXskylxTeHFq2\nAQCQSLYBAEBNlZ+N3AowL+lGDgCAJJJtAABQC1zuvcl2NrORAwAg+WCybTPHBQAA/ifAXFLYGTxn\nGwAAyQeTbQAA4Ccqec62l5ZtAAAkkWwDAIAj5S17STdyAACMGiXbL774otq1a6eQkBCddtppWrFi\nxSG3z8zM1JgxY9SiRQuFhISoU6dOmj9/fqXb0o0cAAA/sXcG8hHLF+n63D/1gfKYIA0AgL3c1d1h\n1qxZGjdunKZOnapevXpp0qRJGjBggH7//XdFRUUdsH1xcbH69++vuLg4vf/++2rRooW2bt2qiIiI\nWvkAAADAIXu7kb/au5+CPlsmOzuHZBsAgL2qnWxPmjRJI0eO1LBhwyRJkydP1ieffKJp06bp3nvv\nPWD71157TRkZGVq+fLkCAsxMpa1btz5o+TRsAwDgZ8p1S7MLCh0MBAAA31GtbuTFxcVatWqV+vXr\nt2+dy+VS//799c0331S6z5w5c3T66adr9OjRiouLU9euXfXUU0/J6/VWuj0AAPAPrr0t23b5MWAk\n2wAASKpmy3ZaWppKSkoUGxtbYX1sbKzWrVtX6T6bNm3SZ599puuuu07z5s3T+vXrNXr0aJWUlOjh\nhx+ueeQAAMBZlcxGbhcWORAIAAC+p9rdyCtj2/a+u9v783q9io2N1dSpU+VyudS9e3f9+eefevbZ\nZytNtsvfHO/QoYNcLpdatmypli1bSpKGDBmiIUOG1EbYAACgFhTNWaig4BBJkl1c7HA0AAD4hmol\n21FRUQoICFBKSkqF9bt27TqgtbtUfHy8AgMDKyTjJ5xwgnbu3CmPxyO3++AhrF+/XuHh4dUJEQAA\n1BWr3Gi0or1JtsfjTCwAAPiYao3ZbtSokXr27KnFixfvW2fbthYvXqw+ffpUus8ZZ5yhDRs2VFi3\nbt06xcfHV55oM0MaAAD+oXyvttKuaSUlzsQCAICPqfZztseOHaupU6dqxowZWrt2rUaNGqW8vDzd\ncMMNkqRhw4bpwQcf3Lf9Lbfcot27d+uOO+7Q+vXr9cknn+ipp57SrbfeWmsfAgAAOG1vsu3lrjkA\nAFINxmxfffXVSktL0/jx45WSkqKTTz5ZCxYsUHR0tCRp+/btFVqsW7VqpYULF+quu+5St27d1LJl\nS911112VPiYMAAD4qXI5ttfjkXWIYWIAADQENaoJR48erdGjR1f63meffXbAut69e2vZsmVVKtvm\nhjgAAH7NuytNVos4p8MAAMBR1e5GDgAAIKniBGnSvjHc3p2pDgQDAIBv8blkm5ZtAAD8xP6P/QwI\nkCR5U3c7EAwAAL7F55LtgzyuGwAA+JhGP63a+2rvnXK3SbZtkm04wGVxEQnAt/hcsk3LNgAA/sGV\nsWfvK1N5uwIbSZJK0tIdiggAAN/hc8k2AADwT66gIEmSN4Ux26hDPG4OgI8i2QYAALVj7+O+qjNm\ne5XrFK1ynXK0IgIAwDE+l2zTjRwAAD+xd4jsKGXoeu3WB7lpkiRvDbqRr7/49tqMDAAAx9XoOdsA\nAACls5pOVoSayC2Xu4lsK0vePZnVLipr3jJ5snLkDg+r7SgBAHCEz7Vsi5ZtAAD8U7FHatRIdmZ2\njXZfe8qwWg4IAADn+F6yDQAA/MIBD1oqKZEKC+VZ82uNyiva8pckyVtQoMy5Xx9ZcAAAOIxkGwAA\n1Aq7pKTG+zZLvEB2sUd7Pvhcfz70sjZccodWuU6RJyOrFiMEAKDu+FyyzQRpAAD4i/3atktKZLVp\nVaOS4u67XpK06fJ7lP/zxn3rt9zwWI2jAwDAST6XbAMAAD/h2i/Z9toKuuBs8zKrei3S7qiIfa+z\nFy7f9zpz9pe0bgMA/JLPJdu0bAMA4Ke8XlnRUZIkz5bt1dvXstT6lYcPXG/b2vHEtHKH8B5JhKii\nnOcmK7XHAOW+NF1ej8fpcADAL/lcsr3/TXIAAOCjKqmz3R3bSZJKfvm9emW5AxQ94jK1nzNp36qQ\nbh0ld4B2PTtT2+55Xin/ekurA3opd8UvRxI1qiD3uSnyrP5ZWWMeUsagG50OBwD8ks8l2wAAwH+5\nTzxeklT0zarq7RceKkmKuPSsfes6//C2wk4/SZK069mZ2j7OJOKpU96vjVBxCFZMlBQUKAUFyrNp\nq9PhAIBf8rlku3w38sTERCUkJCgpKcm5gAAAQJW5u5hkO++FaYfZsnSHAIX2PEFWcHClb8dNuOmA\ndVk8FuzosyW5XHKFBKtk7QaltOml9EuGKa3XJcp5brK8GZlORwgAPs/nku3ykpOTNXv2bA0ZMsTp\nUAAAwP4qGft1sKT5kNwBB32rab9eajqob4V1xTvSlD5rQfWPg2pr9r9XFHjhufL+uVOFcxereMUP\nyr7770pp1lk5T73gdHgA4NN8LtlmgjQAAPxQQIACjmsrSXKFNVZAh3ZV3tW1X9Ie2LZFheXjPnxO\n3QuX6bj5L6jtzL9Lkv68779HFi+qJOi8MxU5b6biPX8o7O/3KPCCvmp8/xgpKFDZD/5DBfM+czpE\nAPBZbqcDAAAAfqpckhzz1/cKiIkqW19S81nDT9z4oTy70iusswID1XTA6ZKknCWrlPbqh0p5/i3F\n3jm0xsfBIVTS+tHk4Tv3vQ69eahSj+2jPZf8Te5OHRQ6bqQa/x89EQGgPJ9r2QYAAH7Ocsk+gkd0\nWZalwLiog75/zJQHJUnZX6zWn+MnK3vp6hofCzXjbtdGkSvmqtFpPeVZt1FZI+5W2jlXKOefL8qb\nvsfp8ADAJ/hesk03cgAA/JvLJVUh2d590XUK9BZWu3jLMpcvnl3p2vn3V/X7WTfx/G0HBJ7STVHL\nZis2d72s+FgVf7Fc2fc9qZTILkpp0UMZN98jzx9/Oh0mADjG95JtAADgd1xWuUsKy6pSsl00/3NF\neVNrfMzcZT/ue706oJdyvvnxEFujeqre+mEFByv2r+8Vm7teEe9OUeCF58rOzlH+K28rtU0v7XC3\n1q4OZyhj2O3K/3iRvB7PYcv05uQo4+Z7VPj1iiP5EADgKJ9LtpkgDQAA/2DvrbRHKUODrhu671Gd\ndnaOvClVT6JdebnVPnb830cdsG5dnxuVnjRf3ryCapeHI2eFhirkyksVOW+m4rJ/V+Sy2QoadIGs\nmCiVbNii/Df/p4yB1yslsK1SYk5SSsse2n3ulcp56gV5tu+oUFbe1LeU/8rbSj/zMuX8a4qKVjBU\nAID/YYI0AABwRCYrQsfNfEtWVHOzovjwLZflNdq2sdrHbPHwCO2YMFXyehVxxXny5uQra8E32nzt\nw/u2ibrpMrWZ+vAhSsEhVfJot+oIPL2nmn/4+r5lb1q6cifPUOHCL+X56TfZ6Rkq2rFLRUu+UfaD\n/5ArNESB556hkJuGyLun7Dne2eMeLyvznNMV+fl7RxQXANQV30u2j+x7HQAAOMwVES47I6vK27uz\nazihlr23q7o7QB3mv6CCjdv0W4/r5M0yLeVpr3yoJhecpuZX9q9Z+Q3ZUehqaEU1V5OH76wwq7nX\n61XR/M9V8M4cFc77TIWfLFLhJ4v2vR/+yj9lp2eoJCVNef+aqqIl3ygl9iQFHNdOQRf0VZMJY2s9\nTgCoLT6XbNONHAAAP3GQSrvRqSeraNFXVSqiRC4FeIpqdPi4h0do599fVdOL+kiSgtsfo+6ZX8hb\nUKCNV96vrE+WavNV92uzJKtJqLrtWigrOLhGx8LRYVmWgi/up+CL+0mSPNt3qOC9j1W8fJXsgkKF\n3Dhk34R4gX1PV+a1Y+RN2yNvarqKl61U7vOvyBXexMmPAAAH5XNjtgEAgH+zoiMl25a36PBJtC1L\nrho+iqTl46PU016pqOsHVjx+cLA6fPy8umV/qaaXniVJ8mbnaXXImfrt1GHK+/H3Gh0PR5+7VbzC\n7rxJzZInq/mHr+9LtCUpJOECxeWsV3zJNsV6/lDghedKliXvtr8ky1Kj3t0djBwADuRzyTYt2wAA\n+InydbZVNg4soE0rSVLJ75uqUITZ72g8ussdFqrj5kxSt92LFXpKZ0lS3spf9Vu3a7Umpr8yF3yz\nb1tvQYFWuU7RKtcpSvl3Uq3H4nd8/ILMsiwzEdvuXxTv3a74km0KOu9Mp8MCgAp8LtkGAAB+4iD5\nWEBbk2znTHy+CkW4TLp9FJ+T7W7eVCesmKGOS19TQESYJMmTmqENF96mH9teKk9GlrI/W7Vv+8y5\nXx+1WAAADYfvJdu+fSMVAADsZR+k0g4Zca0kybs7o8L6wq07tO7sm+TJyDqwJfsoJtulmpzRTSfv\nWaKe9kqduOEDNYqPUvHWnfqpzUD9cevTdRoLAKD+871kGwAA+AXbW5Zsf3fp36WtWyWZLr6uJmHy\nrN9cYfud/5iunK9W66+HXpaqMJ77aApuf4xO+mu+jnnpPnlz81W0+a+yN328C3Wd4BQAwBEj2QYA\nADWyc1P+vteF36yS2raVJk6UJFnxMWbiqnJcwYGSpNSX3pU3JVVS2Zhtp1qTY265St3SP1PsvcMU\nPuB0yeWqcBMBAICa8rlkm5vJAAD4h6ICU2mv0/EKVLFZOWeOJMnOzZO8Xnnz8vZt73KVTaKWet04\nSZJXZeuc4g4PU6unb1eH+S+Yid7oRg4AqAU+l2wDAAD/UqIAFauRWcgw47RD/naFJMnz01pJUtqM\nj0w5E9IAACAASURBVLVr0ttlOy0tnYTM2ZbtStGyLdl2hZsjwGGtWCEtXuz4EBHAl7idDmB/tGwD\nAOB/9lXfv/8uPfGErKZNzfpi0+K99fpHyzZ2B0ie0v18LKFzuWT7UuIP+Iuzz5YKCszrESOkDh2k\nK66Q2rd3Ni7AQbRsAwCAmil3g7x8yuz9zwtlz93er5U4tHcXBYQ33rdc7NrbIu5LCW4Db9n2bN8h\nz2/rZeflH35joNTeG2uSpFdfle67Txo40Ll4AB/gc8m2yyUVOB0EAACoprIE1dqVYip0qSyJdgeo\n8WlddcLy6Yq45Ixye/lWy7arkpbtgo3bHIrGGZ6ffpMkWTFRDkcCv2JZUq9eUkmJ9OefUkCAlJPj\ndFSAo3wu2d62RSp9KmdiYqISEhKUlJTkZEgAAOAwDkiZSxPW0p+2pABz2dFiwv9Jkoo6dD5we19Q\nLpbN10/QL8cN1uqws/THXc+paFe6g4HVrbAJdzkdAnzVl19KYWFSVJQ0eLD04IMmybYs869FCyk8\nXNq2TTr/fOnmm6XnnzdjutPSzP+x336T/vjD6U8CHFU+N2a7oFyzdnJyssLDw50LBgAAHFT5ztau\n/R7MXLht596N7H1buyyTbFvuAElSiceHEuxSLleFbuSeNNME4M0rUOrzSUp78V2FnNRBwZ3aqs2M\nx2RZPtduARx9n38u5eZKhYXShx+af5LUrFnZNs2aSXv2SIsWHbqsRo2koCAz5rtZMzPvgyTFxkrx\n8dKGDVK3blK/ftJJJ0mtWpmEvjasWyetXCm9/755koJlSZ07S7t2mYnevv/eHA+oIZ9LtgEAgH95\nUj8pULbGSBqyd92OFz9SpCTb61X+b5ulEu++lm15SiRJ3v9n77zjoyjaB/7dvZLL5ZK79GZCgnQp\nIk2UpiCgIGBFpCiKXRRFsYtiwfL6qsArKiovNsSOCiJIR3roPZSEkt4uyV2SKzu/Pza5JCShBgi/\nd7+fz35ub3Zmdp6ZzWWfeWaep6SK1+IGtE9aCKXqF5CgvWcdWe9/w7EJU3Am7caZtJuCX5cR99Ez\nhI0aeOEaey7QvNU2PFwuMBovdCsqqXhGNmyAFi3g0CHIzISrrqrMs3s3FBaq1u+cHDXvrl2q8pyb\nC9nZqnV89244eBDmz6/7fkuXqpbxCiRJXaau06kW9bAw+O9/1WXrzZtD69Z11zV/PgwapFrXa3vW\nN2+uPI+Lq7yHJKmTAmazegQFQePG6oSA1QoBAZVH796QmHjiMcvIAJsNTKa689RFWpraJ2YzdO+u\nyg/wyiuwYAF4POokRkgI9OoFLVuqfW2xQGCgegQFgV4Pa9eqkxghIWq51avVuo8dg9BQtW6zWS0T\nEKDWYTar4xYersrZtu3py/A/gqZsa2hoaGhoaJwVz9OGMNx04x9fmiEsCHLyQBEkX/cIgM+yXbFM\n221vgPs5Jaop/oV/rQUBsiwTNX4kEY/cRv5PS8n75k8K/1xNxpsz//8p2xVImtW+QfD11zBypHru\n56cqPKAqabKsfoaGqspbhw7qd4dDveZ2qwpldDQkJEBUlFqHyaQqT6Gh8NVXqlLYuDGMHn1qyp9X\nnTDz3b9lS/WoitFYqQSGhcH116vHiVAUNXygzabWXVAAq1ap7Vy5UlUAjx5VlWq7XZ2E2LJFlb1/\n/8p6TCa1DqNRlbfisFphyRI1j9WqKsUGAxw4oDpzu/tuddLg0kvhwQfVCQIhoEkTtUxamnpvh0Od\nLNi16+R9VdFPer36WXqcdyq9XlV0Q0LU8YmJUdtqsajKr8dTqSjrdLBoEfzzT/U6KnxkVEweVCzn\n93hOPIlRX0iS2tdms/opSeoYGI3qoder/ernB6mpahmTSW1vWVllP1RMXqSmwhVXqHl0OrjsMvVZ\nKCiAHTvUZ/e119Q8ANOnq5MtQUFqPwUFqYfNph4V/RsWph6//AJ79lT2mySp42owqOWiotR23H//\nWXeNpmxraGhoaGhonDUd2VjtuyEyBHJSQFHwHLfPWZRbtoXLQ8Vu79L0PMw26/lo6omRJN/6eEVR\n1Jd/XaXSKZtMhA6/ntDh17PZ1gtR5q6jIg2NeqJiWfXll6tKbk6OqkSVlanKSkGBai2GSkXmTBk3\nDnr2VBXzvn1VC3Bt1tkKvwY63dnd73hkWVWKKrDZYGD5ZFbXrrWXKShQJyTKylRlLjkZ5s6FkhIo\nKoK8PLW/vF71E9Ql68uX115ffLz6+f33p9bmggLVUl9UpFry//5bHTOjEdLTVctx585qvsJCcDpV\ny7vZDP7+qsKekaHuX9+379T8V0gS3HUXNGsG69erCrzHo7bhzTfh2mvVfE6nukS+qEg9dzjUfnE6\n1c9Dh9R+adxY7afAQFUZbd0aBg9WFeT0dDV/UZFavrhYPXJz1fsvXarms9vVNLdbrTM/X22Toqjf\nXS5VXqNRvY/FouZNTFSV6i1b1PYkJ6tlDh06cR/Mn69Oiuh06soIj6f897ueVubo9ZqyraGhoaGh\noXEhURXlGNIwoVon0oghhjREheVLCN/+Sm+xU00rf5kUSqUv8qWtxtI7/StMUVX2fF4ghFLe9vIX\n89A6LNeSXueLI/7/CdGAlvRrVGHWrLqX6yoKbNumKk1btqjKZFUKC2HnTtUy7HarylNxsaoA+vnB\nbbepVueUFFVZFAJmzFD/drt1U5Uom021/MXFqQoR1N/e6bPBZoNHH62e9uGH5/f+Nlvl9+P7/nSp\nUFDXr1cnELp0URXhCsvu5Zer1tdTwWyGESPOvC0hITVXLBzPQw+def11UVioHjEx6rM6Z446kXTl\nlarVedIkdaVDVpb6rBqN6qTML7+o5T0edVIqJ0fNk5enPsP5+epht8OLL6p1KYp6j4oxzMpSV0+U\n1E/owwanbDesACAaGhoaGhoaJ8NBZdxsn5rmVc+EIjBEheJKTUcqt4JVKuJU+8f/d/Rd2K5sjs5k\nIHH8TUQN7HTuG388XoXS7QfYEtYbcwf1JVPS1a5QSLKM+1g2ebMXkP3pLzj+2Ur0pAeIfnb0+Wyx\nxv93TsVSJ8uqEga1K3tBQXVbhiuosKCDunx6zBhVQV+1SlWyK6zDVQkNPXnbNE4Pfbl61q1bZZrF\nUmlx/1+gYhk4qLLfe2/16xUO+epCr1cV6aioU7tf1dUbERHqUU80OGVbQ0NDQ0ND42JBVQI86Kqk\nqNqz8FYuhTQ2isaVmk7j7yerCRWWbSp1bdkg43VDwdq9ABSsT+Z6xw/ntvm1EPPGwxx7bhrevEKK\nFq5VE+tQtnW2QDzZ+Ry680VfWtpz/8GdmYdz/U5cR7No9MXLWHt3Ph9Nr39kzQTSoDifVuROnWDr\n1tqvOZ2qlVGvr1elREPj/yMNYO2HhoaGhoaGxsWIoXzK3lNl7t6nbFcsxVYURPm+Z+MlkXicpWx/\n4osadUXd2JlrUz/3fVfc3hp5zgdRT4+ig2c9HZQNXDL1aSQ/I5bu7WvN22zFp8T+63GsA7tjapmI\n5KdaR7I/mI1j9TbchzPY3+dhipZtpOCPledTjLOjIcU812h4mM3q0uKmTS90SzQ0GjyaZVtDQ0ND\nQ0PjjDAFSOCCqhG3lXJl21PgAKBg/j84Vm3xXT/477nkLNtFDACVzmwkg4w5PpyB4jdWdX0a+8b9\n50eIExD56FAiHx1a53VjVBhR40fCeNVTtKIouFLTcaflkP/9IrKnfAfAvmseBKCDqHQipygKrkPH\nMCbGarG6NU5MxeSHpK000NC42Ghwyrb2O6KhoaGhoXFxIdVy7i5Qw3p5PpmJRBiifDFd7uLqS1Mr\n1HS5ylJtSSdflLGeZVnGlBiLKTGWwKvbEfPag6S99LFP6U6bNIOiv9fjOpYFgOvgMSQ/A1eUrrmQ\nzdbQ0NDQOEdoU6kaGhoaGhoaZ0Rt+rBE9UQ9XqLJBARFk94nYdksDKgevKvmFFWWjUs6GeFVWNl5\nPK5C5zlo+flBH2Qh/sOn0IWojn7SJ35C8crNuA4ew3XwGACizK2GGGuIaBaQhsFFOPGkoaGh0uCU\nbe13XUNDQ0ND4+KlBP8aaQIIpoDiif8CII4jAKTQiExUB0vOw9m+/DF3qt6U7RuSWWi9g6Kdh89x\nq88tEY8PQx8eTONf/0Xit6/TdPF0ol4aQ2CfcsdpLteFbeDxiAaq/P+vo2030NC46ND+ajU0NDQ0\nNDTOimoW6lqCeAok/Cn1fbeg7ufOI5QSzACkJuWQvekwn8oPcnhjDv1LfsSvPOb28taVMXQVj4fk\nyT+wYcgb7Hh8Bkp5LOzzwezZs8+oXMzL99EuaxHBg3sRMqw/Qdd2InbSg/hdegkAiqdhKbffr1h2\noZtw3jjTMb3Y+F+RE/53ZP1fkRMublk1ZVtDQ0NDQ0Oj3jBS00orV1HHXRgA8CJThomgcD8AnB4j\nOz5cDEKw57NVfBk2nt7HZvpKLr9nFhsGv858w83sff4rMueuI2XK78w33MyimLsoTcs957LV+wtf\nhaXyPE4YnArfr1x2oZtw3rgoXuIrlpGfhWX7opCznvhfkfV/RU64uGXVlG0NDQ0NDQ2NsyKB1Bpp\n0RNG1kg7SgxH9QkAlGKizVPX0XZ4G9/1/d+s9517HC4+0z2EgwAAimb+ROZv6nW/2BA6/voCkl6N\n712Wns+ylg/XmzznC0lXvgqggVm2K73WaXv7NDQ0NM4GTdnW0NDQ0NDQOCNcxaoVO5aMGtd0QYGA\nasGuRMLlURW4AJykXOGu5qxFeFWls92EfiTeosa2LsFUrd7S4EiuWv8BS50HGeD+hT7pswBQXKdm\nHb4QFpI67ymrkwUnWgp/Nu1tULKew7L/K3KebdkLcc+LaUzP5r7amJ7be54NF7q9mrKtoaGhoaGh\ncUa4PXVbPne+9gtQfQ93CHnYUT1zK7KuzpcZo82f6358kPvFJ4wWM7l61ydkEkEeNvLyJb5v/jLf\nfv0NAKaoYPxiQ9BZ/Dn85w6+sIzlq+inmRX2JAtvns6ssCfJ3XHMV3dDerGVKsKdnWCiQHuJP7f3\nPFPOq5wV3upl+f+/rPVwz7NBU7bPXdn/pTGtihZnW0NDQ0NDQ6NWhBAUFRXVed2BqiQWln/PIAIX\nCoVAQVkpRSiAwiESSCSFlOAmFOZLLKIjA9M/x3P3KApdZZSgMGT7U/zz2hIOfp9E9O1tKCws9N1H\nFxtA2JA2BDYOY9dHyykpLiJ10U5fHofiwlPmYsm4byhyFINDjfFd8Iu67PzPUZ8waNlTAHg8nmp1\nnw5nWraucsVeN8V4KSgowGQz1+s9z6asx+tRx660FO9plm9I/fv/5p5lZepncfHF0d56KKu1t2He\n82zK/n9tb2BgINIJFFhJiIYVvG/jKrixeyEZWLHb7QQFBV3oJmloaGhoaPxPUlhYiNVqvdDN0NDQ\n0NDQaJCcTF9tcJZtDQ0NDQ0NjYZBYGAgdru9zutbbO3pKQ5yBMgjnnxCiA0vIyJ7N8qxYxwe/Bjb\nNpahx1vNIzlI9Ld/B4D9xbcomTqTsK2L0CfEn1K7/r7jU478udP33UYBEgLDlR24etod2JpG+q7N\nCh+PtWk4Q1Y/e1qynysUlwv0emRZJm3Sp2S+9w0t1s/Cv3kCniNp5LS+5tTqkfXIikfd9614AfB/\nYCT6y5pS9tdy5NBg9AlxGHt1xdih3Wm1sWTuX9hHPUbg1NcJGHXbacsIwFtvweTJld9HjoRp086s\nrgtBu3aQknLq+e+5B95/v3paQYHqQdxcvmqhuBhMJvU4HV56CaZMgc2boXHj0yuroaFxTgkMDDzh\ndU3Z1tDQ0NDQ0KgVSZJOOGNvkXQgIAhwYMKCDpu/Sd2VbbNh+uu/rA0djwEX4VSG5uq+9UNfvcLP\nhB6ZoMBA9Ke4mu3m+U+hKArz+32I40g+IcUCpcxDvzUv1chrlo2YZdMFWynn2rgVb8oRCoY9DDod\nlFWGRguUdVjxonS+HqctCFFQSCAyyBKGLlegi41GjgglcOrrOD+YQdH4Sb6yuU060XrvzwDYx76I\nc9pM+ETdx+5X5f7uSR8SJo5xOhjMZhRkgvzNBJxpvznUWOps3gzt20NpKVxMqxWPHIGQEOjdG5xO\neOQRNb20VP1eWgp2O+zfD9OnwxdfwKxZEBAAXq+69Lsux3dBQRAXB9u2VabJsro3e9cuiI+v3ld+\n5SMaGHhx9aGGhkYDVLarLHkfOvQODAY9w4YNY9iwYReuTRoaGhoaGhq1oP7TLsNAMYGE3n0jIaTC\nfzcDkL48GYCWTw8gItbIrnGfkThuENa2iVWqKHcSppzerjZZlhm46AkAFl1y9wnbKDi/O+Y8aRnk\n9x+OkpmNklU5ySBHRyKZ/RHOEiSjAa/TjTs9Cz0eKFaVU9OtA9E/8zj27xeR+e5XarlvVnPJ22Mp\n6nwNgeuXqpVVEck69XUCJz9LwYix6GKiCHzzWcqWr6Hg9gfB5UYpLUU+HWuqL67zWTjScbvVz/jy\n1Qpe75nXda6ZPBmefx5sNlWxdTrV9nbsCN9/f9LiR/LNSAsX4lech7tEh1fIOIUJly2M2HZhhCTa\n1Izbt8PRo5CbCzt3qpMvFfj7Q0mJeh4UBP/8oyr0mzbBli1q+lnE2dbQ0LgwNDxluwqTn/+Oy7tr\nM3gaGhoaGhoNE1UpKy0Pz5UwcyLcf796SVHYM2MVALG9WxLX7zIaPz7ogrTyPOvalM7+Fc/2PUgB\nZvAzgseLvmkioTuXIldRmNLe+ILMFz9SvygSsjUAeVUKnh9HVatPsRdz+EF1SbaMGQPuSoUYSJ88\nk4y3/otwexDuXTTq1JWw0YNwjbkT50ezKH7tQ0o++wbhciOHBiNHRyJKSvFs24Xu0gSUYxn4j7gZ\n60eTqTcqlO0KJf8E4c3ONUcX7yasXRzL751F2rJ9+NnMSHoZxe0lolMCrTb/QSzgwIyhpAyp1Isk\n6SlrcQX+ilJtzGrjz++KgK5IsoRs1CEEKG4vFAhYDuZ9Vq7+zzAKW2djT84iqnU4zbb+WNlH+/ZB\ndrbqJTg5GQoLoU2bmjcKCKj3vtHQ0Di3NDhlW/NGrqGhoaGhcXEhqB7iq4L83ekAxPW7rPqFnj1h\nxQoApAmvq2kV4Y3qG4nzrmyLYicAISt+xnhFLUpTOQHtm1cpJFDsxYjSMl+SbLXQLm8JjhWbKPpn\nG6LMRcjQ69jT+S6U/Uc4cNszONZux300CwC/pvGUJR/GsWYbYaMHgVlVdB1vTgGDHjnYipKRjfdA\nqu8e7oNHkF1lOKd/idQ4EcuTY/AmHzqpjGkr9pE6dyttHu/Nptfn0eyuruTvTCPy6ibYWkYhlyvX\npcUujEjIFYrl8Xg8kJUFM2eqSub48ere51GjoGVL1crcooWaLyUFjEZ45RXIzIRWrVRLdEGBav19\n6y31ekgIGI0UbD+Ct117okU+AoneQB4hUAh+lLGS7hw+kk0gXiLR8U3BwOptm5IHUx4i7vrWeErd\nmCODCGwchs6ow2D1xxxtpSxXXZHQ7O6u9Jp5d7Xic5q/hH1fFs50O4tu/tiXvgfYmBBOwqC2IEvE\nvzqBS65rpV5MToaPPoING8BigcGDISICgoPVTw0NjYuKBqdsa2hoaGhoaFxcWCnCOXxwtTTF46E4\nJbdm5g0bfIr2+eJ8B14RTnU5sBx4Ykuk9Yar6SA2kjd7AbahfX0WVMXjwZm0G9NllyLLMoG9OhLY\nq6OvnLlLa4qXbqTgx8UAGGLCiX17LNYhvdga2MM3ueA//GbKfl6AGHwjqcamFKcVYo4KwnJJMM3u\n7MCB/mNxJu3GgJtwcnA8/SqOp1/13Ue2VXqi3/XJCo4t3IUzo5DsjSkoLnVZ+PZ//w3gW8VQQWtp\nH12BLyOfYTQ60v/cSrAtGn97hjotY7OpSvLxvPOO+vnWW6qy6XRWTsSEh6tOxuqamKkaF9doxM9r\nxF8Uk0cw9rjL8C/KIlKXh+R0QEkJA5nnyy4kme6fjqA4NQ9LoxBKs4vZNGke3lI3RxbsOOmEjbVZ\nZI2023a/ivNoAbnbjlJ8OA9DkIngFlH80mkyxSm57JiibgnY8cESkODqacOwxIcQO/lt9CbjiW+o\noaFRrygeD64jmXhz7SDLGC6JwBgRctb1NmhlW7Nya2hoaGhoNHwkQJ8YWy3ti+CnAB3mWFv1zF26\nVJ5XXZ57mnu2T4uzqHpevw/RGXTE39iWVg/0OKUySma2ehJoOaX8IcP6V/su6/VYutRtEW++5GMU\nRcGTkYs+KrRSSXeWAurkgqfUxeIXllKo6439/RQgpVod9v3ZmPYeRvj5wdCh2P9ZhnLgEMFdmmPu\n2paSMokNL83F+fRv2PXh2PdmAiDpZIRXVXbbv3gDOUmHyfhnP02Gd1Gt82Ue8vdkYN2zA5Evcemw\nTpTMnk0saVDVsX1BASI0FM+1fXEbzLgi49BlpSMcDnR7diHZC5DcbspMkUilJRhKCjEUOpGQ8GJk\nRfgtGMOt9Hyzv+o4bMsW1bmY2QwzZkBpKQa8pBJPI+8hQo5fCp6Tozo1W70awsKQevSg5aju1bK0\nf+5637miKLgKS8lckYx/tJW8bceQDTJ+IQEEXBJM2OVxNcZJlmUs8SFY4qu/sN++bxLOdDuWuGAK\n9mSyeux3FB7I5p9H1MmCJsM7c+3X99Y69sopLGvX0PhfwbltH7mz5iE8XnVrjVfBuWUvZfuPIjwe\nEAKhCHWCThEIRfHlE+Vp1DUZq9PRwbPurNvYoP9ac49e6BZoaGhoaGhonBL6cmdPxykCd6a8UT1f\nxYtNs2bl+c/tzLokSSAEkydPpnPnzgQFBREZGclNN93Evn37quUtKyvjkUceISwsjMDAQG66cTC7\nFyZxeN52Vj34Db9f8x7vWcfQwdocs8mfqKgoJkyYgKIouHcnk92+LxlBzVg862v6kY0lIZ5mzZox\na9asepdLlmWMMeHIsozi8bDp9XmsfupHAPb9uouvY57hyPwd2JOz+JPNPCjNYNMdHkLaqJMiW6b+\nzevFyfQu20DnL1/g/oKDHMPA9nUOdn2ykmPT/6Bo52omH/iK0Xtf50VmsDngVzrHH+O2X4dxj2Mq\nnV4bjP+ENkxrsorrvriHexe/ys6AXTQr3Ua4JwMJQe9vx2BevZgjT77HtwzjM+7lM8NDfMr9zMi9\nhZk/BPL1tzq+fz+N2d8IvvvVzDd7OvJ1eh++yrme73N6M6d4IF9772Qm9/AF9/Kl/h4OZQeyd5fC\n7Ok76XP3cwS/9CqmWV/ReNbPTGn/KD+2eZWZuvv4S+rPy6+8QkxMDGazmeuuu479+/dDWBhMmAC/\n/kr+u+8y/K+/sFqtBAcHM2bMGBwV3tTL2bFjB30H9afF0CvpdGtPfs9bTbNRXWk0sG2tivaJsDWN\nJKZHM4ISw4m/vjV37H+dUbnv0eMzda/+wR+SWHznZyy8eToLBk5jXt8P2P35SoYmXEekzoZRMhBl\nDOH111+vUffLL79cU9Yq5OfnM3z48BPKum3bNnr06IG/vz+NGjXi3XffPS35TpeVK1cyaNAgYmNj\nkWWZ3377rd7kUhQFxeNBcblOSa4ffviBli1b4u/vT7t27fjzzz/Pi5wej4dnnnmGtm3bYrFYiI2N\n5a677iI9Pf2Ect57zz1nNH7nUs6TyXo8DzzwALIsM2XKlGrptY2p/Ugajg07KV6zjfS3ZvJbuyEM\n/vfLNJsygS5TX+S9j6biWL0NT1YestmEHBTAEj8Ht5RupqtzNcOUnWyIMmBqcykBV7YhsHcnrAO7\nY7utDyF3DSDiyeFEv/oAUc/dTdTzo+ulLxq0ZXvnCrj2jgvdCg0NDQ0NDY2TIVX1rFxO/IA2yPpa\nXjV69gS9Xt2fWo442z3bJ1gqLhCsXLmSsWPH0rFjRzweD8899xx9+/Zl9+7d+Pv7AzBu3Dj+/PNP\nfvrpJ4KCghh98wh2kMfTqI7dji3by7/5GRsBPM1Arpo1hhE3DsL17nSeUQOecQQPo8jjwbvu5ofn\nnuHvv/9mzJgxxMTEcN11152djOWkLduLtVkkkizx65VvUZya55O0A1CU68IlOdGbjXBzLEmL5tEu\nuh3maCs3f/Mi89s9x8yUH9hQnM1nBGNBYmJuMi8DP5KDsaQMM8WMwsUlCNYBacBIRxZBh+Yzccjf\n7InphvH7Nxg4YAAP3f8A73cawPwvvuaxae/wFSEMI8+3oEDftRNxXTsRsP5d3LvTMUdZcReXEtQ0\nAmuTCPyCzej81OdE529E56dHZzIg63UENQ3H1iwSU1gAxxbv9e3/T1uxj+97vsnzf71FC2J4mN5Y\nMJFVaKd4zVHydVbMMTZWxqTw/bQfmDVrFomJibz44ov069eP3bt3YzSqS7XvvPNOMjMzWbx4MS6X\ni7vvvpsHHniAr7/+GoCioiL69etH3759+eSTT9i+fTujR4/2KQD1gSnEQot7r2bNuDm4i8s4MHtD\nteszFn3LErZzN9cQQzAp7mzeeecdAgMsPHDXGEwhFt5++22mTZvW4GU9HofDweWXX84999zDLbfc\nUuP6qcqVumwdH4kEXB4Pr3z+Nbd9/guvoUY+cODlFmknXf0j+C6kC3sL8nh5wrMUvfI5Nxuj0EeE\nsMOiMHLz7zzV9Ep6derIsuaBDBkyhM2bN9OqVatzKqfT6WTLli1MnDiRtm3bkp+fz2OPPcbgwYNZ\nv369L1/V8dve9W5emfktt836jdfkS7EN6YW3ZTzXvf883aIa83v/e9lbmM3Tz7+AsmAddzTviLeg\niC2ObO6c9wVvv/MO/br3ZPa33zJk0CAWjX+DDrcOJKDjqcuqKApKsRNPZi7GRtHI5eNxvKxKmUtd\neWMyVluZ8euvv7J+/XpiY2Nr1H38s3rX8BEMO25MHyWZns3aMPPTqezct5f7xo2l1ZvPcN/YR5Bl\nmTVr1vBsjx68/fbbDBgwgG+//ZbH3nqLzfM206IexvSUEA2MpNVCRGEXgPjwQfuFbo6GhoaGCTNq\n4AAAIABJREFUhoZGHSTJTQQg7CD2v/GtmvjAA0KAmMG94p9x39UsBEIMHixEnz5CSJKwPzdZpBEj\nXLv2nXE7FsbeJRaE3lnrtc8DHhXfNX+pRnp2draQJEmsXLlSCCGE3W4XRqNR/Pzzz0IIIfZ+tUa8\nyu1CAvHFXW8LIYT45t+fC72sEzOaPi0+4X4hhBBvG8OFFUmkXzlAZDbvLh6/oqto07p1tXvdcccd\n4vrrrz+5ILt2CfHQQ0IMGCCEw1HtktfrFfP6fSBmN3lBfML91Y4ZhofE38NmiOTZa8VGOojkYS8I\nIYQoKioSzZo1E4tffFH0uuoqcV+Xa8UeXUthB2EE8TMIL5Io0lnFZn+bkECsBeGWDWI+CD2IvdhE\nKnHiSEAzMVUyCBsIN4h0IsU9hIommMQW2op9NBHHiBa3SnpxvTr1IQraXn1ymc+Cp8c/Jbpf3U3s\n/Hi5SJ2/XdgPZgmv11stT3R0tPj3v//t+26324XJZBJz5swp7/JdQpIksWnTJl+eBQsWCJ1OJ9LT\n04UQQnz00UciNDRUuN1uX55nn31WtGzZst5lcpe5RVlRiU8Or9crUudvF9d26C6GXneTyNl+VCwZ\n9YX4hPvFFSSKK2nqew6smMXL908QB3/dJNY88Z3YMO1PYTKZxGevvifcJWU+Wdf8vbxByFobkiSJ\nuXPnVks71TH8mpZiIx3Erg4jxIzL+gmdJIm1Qx4T+4c+K54zNxU2ySA2WHuITdaeYktUX3GPJVE0\n1lvEJmtPsVHfWVxHsOiOVWyUOoqNdBAb6SDaECBut8SLg3e9LA7dO0kcHP6iSHnwTXH0peni2Kuf\nir3XPij29X9UZH3yk8iY+p04NvFj4di8p1r7vWVlwp1bIEpT0oRje7IoXJ4kJCTx1djnT9gXGzZs\nELIsiyNHjlSTc9OmTcLrdouNdBBTaSp0SGIBbcVGOohniRdWdGItV/hkuJsokYjJ990nZ/n3Cjlv\nJVxspINIMnQRm8N6i63R/cTW6H5ic+i1YrOtl9gU1ENssnQTm8xXiyRjl2rlN9JBbEu4URQn7RY5\nX88X2xJvFNsSbhSbAq4WEoj3uNSXb5Olu0gydRV/J/YXUXp/Ma/TbSIhIUF88MEHwp2dL44+O1X8\n1fdeIYH49eo7xN6+j4g91zwgPo65SuhArO73gDj60nTx2tU3ihCr7YTP6tChQ8WNN95YrV+vvPJK\n8dBDD53GU3l2NGjLtoaGhoaGhsZFQi2WbamuvaV6fU3HLOfKGznUavUuKChAkiRCQtT9tElJSXg8\nHnr37g2AM81OFDZCpEByW6ht3VuQSpt2bYmLTyA1eSsA1+jMPIsg8+M3aNeuHZt79qRPhyoW7G3b\n6Ge388TChWps5cRE+OsveOYZiI/Hm5GFbud2tT+qtHPHjc+S36wLHV69kR9avYLR6k/RwRzf9aBL\nwwlsHEbCkMtpfs9V6E1GFEVh8zAI/G02Xvkt7kPP9cLDta+/zmuABWgOLAE8QJNuw/DM+QBLTASX\nA/EJCawdN44u48axduJE2vz+OyHjJmEb0gt9kIWBKSk81rgxO4WgHZnsBwYC7djma9cAAU8AO7iM\n4EHDqXSxVv/M+3M+/fv355XF01i+fDmxsbE8/PDDPgvsoUOHyMjI8I0pQFBQEF26dGHNmjXcfvvt\nrF27luDgYNq3b+/L06dPHyRJYt26dQwePJi1a9fSo0cP9FVWafTr14933nkHu92O1Vp/UuqNejBW\n3keWZeKvb02fLdczY8YMNk6fjWH1fhyGFA6507lH144oJZNMUUwhTkI+Xc+OT3f7YgM0IYDfJk4j\ncuIy/uYwFvS4n/kV+xdxJE+aQ4hBB4rgt4n/oWdMK5Ys+Y2ru3T1bU9Alrnu2mt555132Pnxb8iH\nC3DsSyOofWOavXB7vcldF6czhi3yzFgH9aDJ3H/TzOvlIZOJjLuvpcvgwRy7K41rilrT8eefffWM\nXLaM//buTePUuVitVvY2asT48ePp8Nhj5Mz8Dftfa+j6UybLirPJmzWvtub5KFywxnee/uqMU5BM\nkDV1DknTFqKzWpCtAaAIDBEhmDtfhuIoYXfaASQBFll9Hqo+q54c1bnggAdH88RnL1M49TGuan0F\nqZMn0qOkGe2//Bp9VCi4PIz4+2++vGkw8bu/J7RJArujYxhlScTa6kr0YTYUl5trdgawOPMQAYlt\nKTt0DKWkDKX8N1n2N4FeRtLp1EOvQzLoMUSHoQsORBdoJu+7hbhS0tjTYYRPQsmgR7b4gwOCrutC\nQIk/rkPHkIwGylIzeC5lNSNFGJEbDuIijaPj3mPruK8A+IccgtBxyepkitgPkkR7BBIS6cN70HXk\nnSSnrqFnxDUn/Ltcs2YN48ePr9bz/fr1Y+7cuacwRvVDg1O2z+X/Wg0NDQ0NDY36Q6eo4ZyKCKhV\nsZb1NRVwAAyGyvN68IYq1RJ2rOrV4xFCMG7cOLp16+ZbHpqRkYHRaCQoKKha3vgWieTa83x5IiMj\nqzU5rNz9TUZGBu3atVPz/PwzTJkCXtVjdyRQCJS1bo1f1cqTkynFTABQrLeRq9hIUeLoKVZwYEkq\nmUvK2P2x6rm9LNeBzqTnli0vYWseVaukcloaEWQS6TjKd8BOFKYQSAb+eMnBo9OjDOxPZrduGF96\niTYrv61WPjIykozMzGqyho0aWO06wI4rb8C0dj9p7OeGJo3h5pvVrQEWC5FpaRQOH06RYiS4riGp\nJw4ePMj06dMZP348L7zwAuvWreOxxx7DZDIxYsQIMjIykCTJ1+5qcmZk+OSMOC6klk6nIyQkpFqe\nxo0b16ij4lp9Ktt18eyzz1JYWMgNb01ABhTgbv+29A9Tn9+SMg9SlkRwaDT+sdEUJmfiKXETkxBH\nSambkKatcGU4sCb7YU/az8p2j/vqDkTP1k/nEsNWDpBEFGbm64b4rh+hCFBY+dAHXILq9C9r/sbz\nomyf1hjmgVSueJ3JGFY88wBhowcRNnoQl0+P57dJk2h/qNzbvtGI52gWnsJiRKkLfXgwotRF8dpt\n6ENtiDI3BXOX4cnKRzKbkE1GZJMfksmI7O+HbDYhmfzgpTEE9uiAf7EB16E0FLsDr70I95FMnEm7\ncaEwmb30w8aBSwZibBTDltw9WEtdHHtpOvY/VqrNCQ4kJCSEPMlDYLf25HrKaNy8CcZLyvtLryem\n6aUA5Ak34bJMVkE+bad8SJOhQ3190WL6dL6fNIkWq784o3GyDuxB3pyFmJrGYYyLwr9tEwI6lYd8\nlGUiHh1Ki0GDfPknT55M6IoVvPnDT6Te9QrygmTMLVsS1vkaDLER+EkZRH/1FR12b6x2n5DISLKd\nRWc0plXzVDwX54MGp2xX5TxH6tDQ0NDQ0NA4DWRUZbKQIPwMNV8pJF0VrXTVKjU2MqjK9vFK9rn8\np39c1Q8//DC7du1i1apVteencg+5QKhO1k6ST5o5U/WCfZzTNXr0QAwYAM88g3TllXgbNyGz2MyR\nDAPJW504ywxqA93qPaKDnZAPTbqGk7VBRngUjFZ/bt32cg2v1oDqhfvwYfj+e/j1V+JwkKozMM4W\nxDe3P8Lh6fO47OBcdPeMQN++PfK//109RFZVWcSJZa3A9thdmNrsQ/zxCYwZo1rpK+qYP1/tj5PW\ncvYoikLnzp157bXXAGjXrh07d+5k+vTpjBgxos5ypyLnyfKI8uf1VPqrPpgzZw7ffvstz/h1oUl0\nPH6vD+Lxxx+n1xvjGTlyJAFr1kC3P+m3830iIyP5ree/yP1nP9ZOTdHr9Vz17VssnzwZ/48Pcck1\n16IPMhPWux1B7Ruja9OM+Nv70q7b9ZheSMbg0BHZozOGYAtCUSgryIA/lnPZO6Np1787m4a+i/NQ\n5nmRuwJHSibbJ/2CrXNT4u7qXff4GCon+E5lDE8ljyRJyCaTL80YH8XxgdlMzRv5zoNvufbkAr00\nhsgnh9OqigKqKApKYTEel5vb7hpJQJbExMJY5Cw7nvxCvEUOBB4yXv9cLSBLhD98G+KzSfUm55li\nG9QD26BTi9aQlJTElClT2Lx5M3qLmUt/egd94g+EjhpAo8ceA0CaPPmM2nk+ZD1dGrSyraGhoaGh\nodFw0Zcr26CGhDoeSa+DrCyIj4eysioFqywjP8feyJGq69qPPvoo8+fPZ+XKlcTExPjSo6KicLlc\nFBYWVrNu5+Tn+SwjUVFRbNiwARIkgskFWUYqV7qi5sxRP4HMJk1g7161Alkm67//Jchmo3D69/ze\n81+4C0sBN5Lej8Sb29H1w6GYQgJUh2bLlsE1X3PZLS25bHX15Y8+brgB6vIe3KYNW157jeybb+b6\nzyYj9AKaJeD1elmxYgXTpk1jwYIFlJWV1ZA1KyurpqxVyCyfLIm7rAWNht1ObM9FvrSqdQRZLBgK\nz33Am+joaFq2bFktrWXLlvxcvlQ4KioKIQSZmZnVrFtZWVm+ZeNRUVFkZWVVq8Pr9ZKfn09UVJQv\nT21yAjWsZueKCRMm8PzzzxP/xCICAiPoOXw4KSkpTJ48mZEjR56yrLnFdi7/7zjfda/Xi91ZTMsb\nexA3qDdNl3SmqKiITj+/6MtjX7YM5r1H2/sGEWS1Ihv15zZUXxUq5Frz1MdYVx3lyOcG4u7qXccY\nJvocMp7qGFa1mteV53yNsSzLKBYzw267jaMZ6SxZsoTg4Mr1IR1mzmT2E0/Sctk3mJo1QjabLko5\nV61aRXZ2NnFxlV78vV4vTz75JB988AEHDx48q7/LhiQrNPDQX5phW0NDQ0NDo+GilL9GCEDU4nVc\n1klqiK+qijaA8Xi7UD14I68DCXxW80cffZS5c+eydOlS4uPjq+Xr0KEDer2exYsXlzcIMingWGYa\nV111FQBdu3Zl+/bt2MuKiSQThOB3yY8gSablm29CUhJdn36axWazGgKtfGn9woUL6dq1KwsGTMVd\nWIo5xsot21/mPvd0rvvpQSyXBKuKNoBf+ULz3FwYOxZGj4auXSE4GJo2VUNWVSjaQUHqccUVsHIl\npKfDtm306dOH7du3s2XLFrZu3crWrVvp2LEjI0aM8J0bDIZKWYF9+/Zx+PDhGrLm5FTuE1+4cCFW\nq9Wn4Hbt2rVaHRV5Ordrz/ng6quvZm/FpEY5e/fupVEj1cqYmJhIVFRUtTYWFhaybt26anIWFBSw\nefNmX57FixcjhKBz586+PCtWrMDrrZxcWrhwIc2bNz8vS8gVRcFhL8KedACl1I1sVLdhyLLs21db\nQ1YJShTXOZFV0ul8lv2T4Sl14UzNJH/jfgp3pPjkOfLlEnY/N4usP5PqLOvKKSTa30ZkeARrU3cB\nILxKnWO4BydS+QqbM5Grtud50aJFdO3a9ZRkPVs8Hg+33XYbBw8eZPHixdUU7Yr2FRQVslc4kc2q\npf1ilHPUqFFs27bN99u0detWYmJimDBhAn/99ZevjfXxrF5oWYGG54184z+V3sg/ekLzRq6hoaGh\nodFQ2UIjAYidxIiUb1eoiQ8+6PNGnnrVUNX7+PHHiy8K0b+/EJIkstr3FWnEiMJX3jvjdiyKvbtO\nb+RfBD4mZl/6gnjooYeEzWYTK1asEBkZGb6jpKTEl/ehhx4SCQkJYunSpeLrhz4QlxIpurTr6Lvu\n9XpF27ZtRcfIVuJL2ogFIMKQxZPhCb48hw4dEgEBAWLChAliz5494j//+Y8wGAxi0aJFYlbYk+Jz\ny9gTC5OUpPZRo0a1911AgBCtWwtRVnZafdSrVy/xxBNP1Crrxo0bxVVXXSW6detWQ9b+/fuLrVu3\nigULFoiIiAjx4osvnlTWP2bMEhvpII6++NFptbEq7iKHSJ78g8hbvbvW69vHfiJ+uO91YdTpxZtv\nvin2798vvvnmG2GxWMTs2bN9+d5++20REhIifvvtN7Ft2zYxePBg0aRJE1FWpf+uv/560aFDB7F+\n/XqxatUq0axZMzFixAjfdbvdLqKjo8WoUaPEzp07xXfffScCAgLEZ599dsby1ZC3pExk/b1ZbHt4\nuljR8QmxMGqkmB94u5hvvlX8zo2iN3EiDJOYSGcxt99T4ueffxbh4eHiueeeq1XWDzs8KdqRcE5k\nXdllvPhdHiTyNiSLtJ9Xi72vzhYbbn5T/NPjWbEgeJiYZ7pZ/KEfIn6XB4nfufGkx8/mm8RUv95i\niv4aIYEYw2ViKt3El/QUf9FXjKGJCEQvXqeN+IIO4tqgWBEn+4sd/R8R+wY8Lvb0vE/0im8uWmIW\nc4c8cMZyrV69WhgMBvHee++JPXv2iIkTJwo/Pz+xc+fOehnj4uJisWXLFrF582YhSZJ4//33xZYt\nW8Thw4eFx+MRgwYNEvHx8WLbtm3VfqNcLle9jt+5lvNkstZGQkKC+PDDD6ulXSyynowGrWwv+VFT\ntjU0NDQ0NBoqVZXtw7+roZM8997nU7bzY1tVKonPP195vm+fEH37quGh7n5cpBEj0oPPPLTQyZTt\nbxs/LyRJErIs1zhmzZrly1taWioeffRRERoaKsxGf3EFiWL7/HXV6ju8b5/oZQoRASAiQDyss4qM\nNr2r5Vm2bJm44oorhMlkEk2aNBFffvmlEEKIL6OeErMixp9YmB071D7S69XP/HwhvF4hsrPPoGcq\nueaaa6op21VltVgs4tZbbxWZmZnVZT18WAwYMEAEBASIiIgIMWHChBqhtWqT1bljfw1l2+0oEcfm\nrBQb73hHLG39iPjTdof4wzBELL9inPjTOrSaYrYwelTtiplusPhdqq7ATaSzuKxpc+Hv7y9atWol\nPv/88xqyT5w4UURHRwt/f3/Rt29fkZycXO16fn6+GD58uAgKChI2m02MGTNGOI4LvbZt2zbRo0cP\n4e/vL+Li4sS77757xmNRcixHbB/7iVgYPUr8YbxJ/GEYUk2mP3SDxYKQO8WiuNFiSfMHxbK2Y8WK\nwa+Ke/veIuKjYoXZbBZNmjQRL7/8crWwR1VlNcoG0YpLzomsq3s9V/v4yIPEPP9bxJIWD4qVXcaL\nf7o/I9a1vkdsvPQ2sTakv1hOD7GCbmKtfw+xQeooNhi7irXSlWI6LYUEQj7uuJFQX7io+4gWYRiE\nH7K4kiDxC5dVCzu1lHbiekJEoMn/rMbwxx9/FM2bNxcmk0m0adNGLFiw4IzH+XiWLVtW6+/Q6NGj\nRUpKSo1rFd+XL68M1Zafny+GDR161s/quZTzZLLWRmJiYg1lu77+Ls+1rCdDEqJhuSFLWg0Dry4k\nAyuLf7Bz7a1BJy+koaGhoaGhcd7ZKiVwOan0woS+a1fuGXsfkXfO4FqW8jn3cC/lnm2LisBiUfdp\nX3MNLFkCcXFw9CjK2LFkTv0JgGhx7PQb0bEjJCVxWJ9IvPtgjcszrY9jCrMw7MAbp1XtptfnsfGl\n3xiy/jkiOiVUXrjvPvjsM44QS9y375I++gX0LZsQvnnhSev8MvIpJFliZPq7dWc6cACaNFHPAwOh\nsPC02t0QyJyzhKN3TMAZFocnLIrSIzl4HZVbCSSjHuHyVBaQQG+zIBv1uLLt6C0m/BtFYAwLonj3\nUQKaxVCSkoUhLAiDzYzeGoDe7Ac6mWNfLqXrysmEdrvsvMmneDw4ktM5+tVSCpP243GU4cqyI+lk\nXHlFhPW5nCu+qb7f3lVQTN6q3djX7SVl+p+4c4t8fWEMDSSkx2XYOjQhpFdrgjs1O+s2/n7Ne2Ss\nSOY+78dnXdfxFO89RuonfyJ7XHi27UT2uNHJAk9GDoqjFH14MO70bBRHCYqjVC0ky3iFhGzQo/M3\ngMeDzhqIbPFH8iv31F3urVsfEYwxOhxDXCR+TePwb56APiaMDQMmkb1oKzd4fuHozEXYtxwi8oZO\n6CnDGBKIuUNL357tM8FT7KQ0PZ+yrELcOYW4cgpx5RZSlpGPp8CJ2+7AY3fgLizBW1yK11mKt8SF\nJMt03/wBpqia/vcVl4uyLLWusmw77rwi3HnFuPKK8dgdeAqduO1OvEUl6nmBA1dOIV5nGcLjRXgU\nhFdBeL3V9tcaw63YOjXF4yjB6yjDW+JCKXUR3u8KhNdL6dFcLK3iaPXO6DPuD8XlUvsjo4CyzAIi\nbuhwVv17sVCaV0zB7gxc9hLib2hz1vU17B7TwoBpaGhoaGg0WKTyt79phGJ7/SNir23BkjvVGLPd\nW+fBjvKMFjVcUDWP49OmwZAhyP7+SGG1eNk+FXr1giR1z6fNk1t3vjOwKwilwtv0cRdSU/Gg408G\ncP+wYTD6+VMOXya8CnLVsGe1kZgIffqoobSef/60232hUFwudjz+Gce+XIpwOokESnMcuIsy8YsN\nJahdIuF9Lifq5q4+pcRVUIzi8mCKsJ3RPQ98MJdjXy5FKXPXoySVuHIKSbrtLQo2HsDrKKnbmZAs\nIcmy6ndAEaR9u5yytFycKVmUHslBeGu+0Aa2TaDn1innpN3AaYXU8xQWg8eLPsSKUlpazfN2BTmf\nzyXn87m4DmfgLShCKSmrHq9Xln3fPTkF6GwWDDERCI+HxG9ex9K17dmLZNCDEMw33gzlfZo6TfV+\n32ney1jKFUHF5cK+NYWy9HxS/zOPkrQ8lBIXSqkbV14RflHBCLcHb6kbpdSF1+k69djDsoSkk5EN\neqRyR3GeQierOo9HeLy484pUBVkRp/+7o5OR9Tp0AX4Ygi3oTEZkkwGdvxF9kBldgAlJJ5P+wz+4\nsu1kzd/oa4+k06GUukj9aL6vuqx5Gzn86V8oZW78IoPxFJUQd09vrFdcSurHC3BlF+J1lOJxliHc\nHoRXINweFLe31v5o/PRNZ6y8e5wucrceJW35Xgr2ZBDcKoa4/pdRnJJL4aEcig/nEXdDay7p3fLk\nldUDHqfLd1/HsXycaXbyth/j0I+bfHkkvcx97ulnfa8zUrb/85//8K9//csXU3Lq1Kl06tTppOW+\n++477rzzToYMGeLzFHkiGpbNXUNDQ0NDQ+NEZG9M8Z1fuuM39eTjOixrrVurn4qCZPJDOEsA8BxN\nRw4KQA46ycq2b76B5cvBzw+7yx+XVNPpGqihmc7kfcJxNF89qRo/XFFg505EVf+ygmpxrjwuD8vv\n+i+H529n+LG3MVpUxSV9VTLeEnelI7S6kGVYtOj0G3yByF+zh6Tb3qY0LReEaqmNGtAR5s2j1fOD\niX3jkTrLGm2Ws7q3rtxRmFLmOUnOU+fo10s58K9fKdp6yJdmCLbgf1k8ZWn5BDSLwXRJKObGUbhz\ni7j06ZuxNI/15d3x2KekTP2D3OU7kY16BKAz+xExqDMhV7Ug5OqWBF7eGLmWuPT1jVkpZl+fh0j4\n+jX0ESHgcpE941dKNu2l7MBRXEczcaWk13jhlowG1Yt/LUqXLjgIQ0w4hqhQjPFRBPXvSuiIG865\nLAC2ri3I/msThuBALC0vwV3gQOdvpGDdPjYMmIQxworH7qx18kUX6K+eKAJ3bpGqMPsbMdgCkGQJ\n4VUIaBZLYOt4jCGBGEICAfBPjMDSLAZjuBW9ueYkhH3bIVa2H0fpkRyQZfSBJgJbN0If4Icu0Iw+\n0B+DLQC91Ywh2KIeIYH4hQdhDAvCL9KK3mI+rX7Yet9UirancuWS16u1qWjnYVwFDgKbx5L3z262\nP/gR6CSUMjclh7NBCA7+61dffsmgQ2fxR+dvRA7wQzLo0QWYMFS01RaAITQQXYCJ/a/NwZ1fXK0d\nnlIXJZlF2Pdnkb8jjdwtR3AcKyB9RbLap4rwWeZrY/0z1XXB/d+sY2TGv06rL04VRVFw5TnZ/90G\nVo/97oR5w7skkjC4HYGNw+rl3qetbM+ZM4fx48fz6aef0rlzZ95//3369evHvn37CAuru1Gpqak8\n/fTT9OhxajHYAIRm2dbQ0NDQ0GjwCCSSXv2Dlvd396XpKpanPfBA7YUqlA1FAZ0OvApKYSHZcR2R\n42KIPLyh9nIVTJumfmZlIWwJJ2vgaZG+KpnD81WzfGB8CMyZA+++C9u3g8uFSxdIZdQzQeYBB79I\nD6hWJknyvVwWpeSi9zcw96p3KM1Slw37RwaeXmMaGK6cQjaP/DfFu49SlpmPUqoqNuYm0YT1aUfb\n6Q9TsvsQu+bNO+dtkY1qTOXTsWwf/Xop1g5NCGwZVy3d4yxlUcTIasvdA5rGEP9APy4df9Mp1996\nyv20eOdu9KbqkyqKouBJy6Zk236yp6zD1LIx1n5n5xG5os6iZUkU/r2OvFnzMLVpgic7n8iMAmLw\nUrQYtkf3r1lYlpH9/fBrHItfiwSKV25GF+CPOz0HQ2w4uiALOpsFXYgVSa9DH2Yj+uUxGKPqRwE5\nE5q9cDvNXri9Rvryyx/HsU/dghLQLAb/RhEEtU3AFB+GX4SNyAEdkGuJgFAfWNsmMtA795zUXRft\nZoytNT3wssoIC1GDuxA1uEu16x5nKVl/bMTjKMUUE0JEvytq1FGUmstv3d+lZG0aim+rhyAG2P7Z\nejZ89cgJFWhfWxqHYQw0obf4YQqzYAqzYIkLJrxTAhGdE9n2/t9IsoQ52kpgYhgr7v0ST0nNv2PF\n4yF13g5cBU48ThfeEjeeEjcuuxNXfgllBU48Dpe6hcPuxFVQgquoFI+jDG+JG6/Lg+JRaoSqS7jp\ncmzNo/CPCiIg1kZAXDCBjUIxRQTW+0TYaSvb77//Pg888ACjRo0C4OOPP2bevHl88cUXTJgwodYy\niqIwYsQIJk2axIoVK7Db7XXWX3XVyzmKAqKhoaGhoaFRD0hVtNiMFcmU5TkIPdXCFS80QiDpdSgl\nJRSMVve5Kjl5dZdzOqF7d9i0CaKj1dBX5fXU0Ujq0rZXPPg1h75PYui+1zCGmNnxwWKSXvkDd5G6\nzzSsQyNMYRa44w61QFwc3HYbG/aEw/xyy6eAEmf5C4siCGoegcHiR07SYX5qO6nGra+YOPAEndLw\nOTTld7IXqEstDWFBBLVNJP6+64gf0++8t0WqsGy7ar6kZy/ewqEPfsd5MEPdc1tcWm3Akz9gAAAg\nAElEQVSfeFjvtjhTs9U9tAUO38u4MdxKjx1Tz2hpe+neVNJe/RTXkUw82fmIUheKsxSvvRhxfBv1\nOtrlLsaxYReejFzK9h/BsX4nZXtTiX7lfqz9ryLzw9kUr9qCM2k3otSF8dJLkM1+uI9k4ckpqPWZ\nL92+H11wECVBEWQXmuh0Xzs8uYUIlxvJZCSoT2eCh/VDH3R2qwoaEj23fHihm3BRoDebiLm92wnz\npPy6BceRfCzxIYS0vQRjsD+uwhI8czPws/rhFxeB0eqPKTQAv5AA/IIDCLjERmCjUGL7tfKt5DkZ\nnd8YclzbjJTmOTi6eDfpy5NxHMmjJLOI9BX78Dhcp1SnpJOQDTp0fgZ0/gb8o4IwBJnws5kxhVow\nhQfiHxlIeMdG9bIX+1Q5LWXb7XaTlJTE81X2EEmSRJ8+fVizZk2d5V599VUiIiIYPXo0K1asOOX7\nKd6T59HQ0NDQ0NC4QOgNcKIVvE2b1n2tYnZdCLwHUgFw/7MeAN0l0bWXmT0b7rxTPffzg2rxU+va\noyqpCu8TT8Cnn8K6ddC6NYtu/ZhDP6kxXA/+tInNb8zHcURdOq7zNxDSOpbrFz6uVmE0QvPmsG0b\nAJ7bPwUqlhmrCo9faADD095Bb9Rz4PuNLB46A4PFRMy1zen0xhBkvYxQBMEt65DtIsFTrC7377nr\nPzWsw+cb2U99jfU6XWTO30j2X5sRipfDn/ylLoEGZJMRQ4gFqcRVbd4jZ/E2dAF+6AP9Ce7aAr+o\nYAJbx3PphJtrXS58MlIffouc6T+qX3RyebxnCZ01AFOrRIyNojE2isavaRzZU76jLPkIW629aq0r\nZeTL1eUMNCPcHsr2pCAZ9MiBZpAkrIN7YoyLRPb3I2RYP0zNGoHJiCzL/NHnffKW7qXR/7F33/Ft\nlWfDx3/asry3YzuJ7ewdCCPsFWaZhQbCHmWlUJ5CXyhPKQQKlIfSMjqg7EIp0EJbRghpCSRQwggJ\nJIRsZ3rFe8raev+4JMs7dmJbcnJ9Px9Z0pn3OTqWznXPp+/qNd0Bn4+W0gYatlTSVFxF4fcPxp6R\ngKfZRcOGCuo37qZ61U5adtXhrGxkwlVHMuGKI/t9ftTw4HNKYHvSa9eSfUQRINfIe5bFjDtvGjNf\nuGVQ9mtyWAi4fbw357HIRAMYzSbsWYkc9buLMDus8oizYsuIxzEiuc/BfV/5nC4+nnEL3pomTA4b\nc0pe2Odt9ivYrq6uxu/3k52d3WF6dnY2Gzdu7HadTz/9lBdeeIHVq1f3O3FajVwppZSKXb5AxwC3\naWs1I0z+SBXrs87qeeV21ciNeTkESisI7K7ufYfhQBukl+52VUMNPdUVN0AwGITHQjdxy5fD1Kns\nXLi2bZH/3vAKAJmHFXDOp/+va4+7wSDExfWYLK9PSlTMVllvzNxDGH329C5VifcH/lC1cVt2cpRT\nAsZQyfaaq7t2NOYoymHWmz8jeWZR27SAx4OrogFvdQOJ0wv2qmdlX7MT745yGhb+F+c3m3Bv3oW3\nrApvWRUAqRefRtEr9/e6DVtBLuULniZu+jjskwqw5GZisFlIOftYyn/5HM2frsaSnYatKI8R91yL\n0WrF19iM0WzGuBcZAWEBn4/G4mpqVpfwyXV/wdviJujreLP9yfWv9LqNpq3VGmzvx7zN0ozCktTu\nOgt9Vwf7WOXY53TRtK6EsleW4nd7mfzwFXtsl37CS1ez+aXPictJIn1GPiOOG7dX35+BQABXSQ3O\nrRX4ml24y2rxt7oxWsz4PV7c5XV4KhvwVDfirWvG29CCp6oRX0OLVJsP1XAxxllxFOX0e//dGZDe\nyIPBIIZuej1sbm7msssu45lnniE1tWt3+Hty8S3jsP3MQF5eHnl50vnEvHnzmDdv3j6nWSmllFL7\nKhj6K/cAfrePYLDd/UA3vRq3aVeN3P79M3D+7nkMqcnQTZXgLmpqOgTaPZdqdzMnGGTlve/gd3nJ\nOqKIys9kuLD0g0fyvSU/kQAsEJD0uVxw663g9UpJelvaZatPG67nRKz4MGFJ7His+0Og7SqroeQv\nS6l8dwUtWytwl9W2VYs3D3CJ0t7IPHkGSQePwZaZRMZJM8g4eSa2rGSsOandtrs0Wq04RmXCqMw+\n72PTyfNpWbGOoMsjVcE7V902mzAlxWOfVMiIBdeRNvfkPW4z5cxjSDnzmG7n5f3yxm6n97vatwEI\nBPnbpHtw1bbgd3raAqmwxKIMMg4eRUJBOo6sRKpX7aJ+Y4W0rx2VRmNxFbknTGDsJYeRODqNl3Nu\nb+ulHySwadpWzc53v6Vq5Q6aiqtp3lVL/pxJ+H1+nGUNNG+vkWrILR58Tg/ZR43BXevEXddC9uxC\nTvnn/P4dl+pV/ebdbHtzFU3F1VgS7Rzx2x90Wabs40188sOX8Ta78bul9/GA108gNNQYgLXd91n4\nf6np2x1s+907tJbU4K6IBK2+hhZ8Ta3STtrpbuspPix+fC5j/uecXtOdMXMkGTO7rynjqqijYVUx\nzuJyAl4/rpJqWndV0/TtDrx1zfhb3AQ8XoI+f7/65zCYjBgsJsxJDhxjR2DLTMac5MA+Mp0pj183\nYG23+xVsZ2RkYDKZ2L17d4fplZWVXUq7AYqLi9mxYwdnnXUW4eG8A6FcEavVysaNGyksLOxxf39+\nZDNnX6vjbCullFIxqdONjcFsJCHBBOFRuHopDSZcqhgMYr/wTFxvLCRtyevUHPa9nteJi4OiIkjr\nx1BhBkOHTmA+vPPfbKmTTp6OePQHZB5a0PGm6s034YILZL1wYJWWBrffTiAQoOX+x0n/fBlbidz3\npKYaOHrdPX1PUwwLBALsevY/bLrnFdwV9R3mGcwmuaGFQetwqj/MCQ6OXflov9cLuFwEnG7MaT2X\nznt2VlC/8BOaPvwKg9WCbfxILDkZWPOyALDkZzHi51d1O0xWLMg6tICyJRtp2lGDJcGGNdVB9lFj\nSChIJ3lMJrknTCDzkIJ+bdNgMOAsreeFpFvwtXq6lIpjNEAgyMYXlrdNMsdbsSTKkFUAu5dvxWQ3\nE3D72LV43b4eJiDXbN26curXlbP55S9o3FpFMBAg6/BCTnhx78eZjhVVq3ayZO7TBHx+vM3uyNjb\nyHCCwUAQAsHI63Y2vfQZJrsFv9MjmaGBYFvtFHtmAmaHFZPNLNWzE2xt6zvyu/ZZ0Pj1Vr77emtk\ngkG+E4w2CyaHDVtmMpbUBBKmjKTs1Y/bmnK073QwEAi0fd8GAoG2kueWzeVUvLEc5/bduMpr8VQ3\n4W9qlf4Yeqq0ZDZiSU/CnpuKOSUBa0YSlpR4bNkp2HPTMMXbsWWnYEqKw9/swpRgxzEqC/vozCHN\nDO1XsG2xWJg1axZLlizh7LPPBqRUe8mSJfz4xz/usvykSZP49ttvO0z7+c9/TnNzM0888QQjR3af\ngxE+p35ts62UUkrFrK6lxp2mffAB3NVDm9FwgOv3Yzv6cLLLVnW/XIftBzsOxRWebICkQD2MHg0L\nF0JWliz30ktcVPsHvERurJrqIiXnqVNyI4H2zp1QUBAJsINBKZl//HG47joC9Q1UJo4n6GwlBzgd\nM01nXohjkZuUyal7VSU5WlpLqqlZ+i0V//yc6iWr5Qa8qRVrZjLeumYJqA0GMk+ZyajrTiNx6mgS\nJuTRvLGUpRO7L3ntVgyN4Vr7+mJKfvo4vuoGgq7Qzb/RiGPWROwTC/A3NNPy5Xf4axu7dGaWdNoR\njP3n4AxJNFgOe/A8Dnuw772o90XeKZMpWbQWa6qDuKxE4nNTSChIJ3VyLkUXzsKaYG8b/i91am6v\nAc0b0++jYfPuLtN9Tg8VnxVT8ckWGjbtpmGTLOOsaMTX7MbssLaVyEKw256mATAaaNxSFXPBdiAQ\noHFzJTWrS6hfX07zjloC/kDbF2fr7iZqvtnVVuIc9AXagmOjxYQ1JQ5LqgOjyQjGUIdgVhNGqxmT\nzYwjJ5lxl88mLjuJd457BH+oh+9wr+BBwJ4eT+6JE7t0UtabWf+4E3dlPY5RWTiKcnAUZvaa4Tbh\nvovZ/sdFbH34H5S8uISGFZtxbttN05rtkpFpNHQpAZeDNGCyWzEnO7CPz8Wen4GjMJuEiXnEj83F\nYDXhKMohbnTWkAyfNxD6/ctw6623csUVVzBr1qy2ob+cTidXXnklAJdffjn5+fk8+OCDWK1WJk+e\n3GH9lJQUDAYDkyZ1P2i5wQDO0OteOi1XSimlVLSF49LQnWLQ78doaXcDNGVKz+u2q0beL900W0sO\n1GEkKAHztI69zNoAG5HebGfOHc/o17sJnB56KBJgP/UUXHZZWxpb31xI/aU3g8uNeeoEAg1NmHeV\nkZvmprWHNMWius82sOqS39C6rWuQA+BtaCFhUj6Zp89i4i8v7nIznTAhj/wrT9pjj8axaNslv2i7\nubdPHYNrbTEEAjhXrMO5IlLCai3Kwz6xgKDPj6+iBmNCHLn3XhetZMeUk165Zo/L9LW03Gg14Xf5\neNp0A/b0ePytXrwt7l6rAZvsFowWE5YEG/H5qZisJqypDqxJdlIn55I8PoukcVlkzy7knRMebWsi\n0h1Ps4vKL7bRVFxF045aXJVNTL/9FGpXl1D82gpGHD+ByTccQ+O2Ghq3VNGyq5YRx43HkZfS1imX\np9FJ5Yod1K0to2FTJVUrd+Cpa8Fks+D3+nDXOvG7vAR9Aamm7e8hY6DzubGYcIxIxmQzY7JLz9oZ\nB4/imCcv6dO5DbvG+ft+Ld+bEef1b5g6x+hsCm8+k60P/wPnlnKcW8oxWM0YzCbs+enE5adjzU7F\nnpuG0W7BnBxP9lmHkjy95xrPw1W/g+25c+dSXV3N3Xffze7du5k5cyaLFy8mM1Pav5SUlGDex9zd\ncH5ii7PXxZRSSikVRUZjENoVTgT9QewJlsiER3up4ttTsG3oZlrbDrov2W40pJASrOt2la9Nh3CQ\n/yuKKWIMWxl9XFG3y+EM3XQ0NUWquAMN/+8+nI/8CYCk539D/FUyDFi5IS+SziEMtuu+2Ej8pJFY\nk3rvcAigeXMpn59wF57aJqzpibhKpH5/0sFjyLvoGHIuOLKthKh99c7eDFZvxIMuEMTosDOj5oO2\nqt++xmbw+PBUVGOfXDRsSsr2B7MWnMXKBe9Qt66coD9IXHYSSUl24rKTyD1hAiOOG0f6jHyMoU4H\n+/vZGIwy3v1/LngKZ3kjrVVNBDx+Al4f7jpnW4lvexue/W/b65LF61h5zzu4a1u62XhkJIWO02UI\nq4DHLx0mJtiwpTgwOSxYE+MwOSzEZSSQUJhByoRs0qblkT4tH6M99H0Tau4ynGrJ9CYuP4MTtvwJ\ng8WMPT/9gP3/2qtPc/78+cyf332HBh9++GGv677wQt+7UPf3NpyIUkoppaLKaLdEqqOF2JPalYb2\n1q63XW/kfRYMdhvYfpV3Dn6nm1Nr/ioT1q6F996D22/n6/ibWeE8mEkzrYz55vc9788dqlrc7ka3\n9swrcC/8AENKEsnP/Jq4CzqOkR30B4BgW4dpg61y8Sq+PG0BcUXZnFT8TI/LNazZxsY7X6Zy0cq2\ngMBVUoM5JZ4jPnqgQw/dYfv9jbDRSNzM8R3aWIc7HTNn9H9MbbVvRp85ndFnTh+07adMyKbi481s\ne/NrDCYjRquJYCCIJcFG4uh0EgrSGXHcOFLGZZM0LotV975LzZoSbKnxNBZXhYJsN4mF6Uz7yRya\ndtbiLKnH09hKwOvH7/ZhtJrIO3Ei6TPySZ85kvjcfbyO9sP/wfgxw3uow4EQ01kn330V6RBUKaWU\nUrHFYLN1CbZtiaEA22TqfeW9rUbe001B+yB86lR5AKctupmGTbuZlNcMZ/RSrfIf/+jwtv7K/8G9\n8APMUyeQvvqDboNR37pNUjXZ7ekybzBUvCkdTwXdXUsjaj5dz9r5T9L83c5QJgDYctPI+f5s0o+Z\nii03lfSje6nWP4AMMXnj1n1Gjdo/Hfv0ZRz79GV9Xv7kN29oe/3WUQ+ze3kxAGMvOZypN5844OlT\nB46YDraX/RuefwB++Itop0QppZRSXXUc+gsgITteXuwpsOkx2Db0Xo28nwFT7rHjyT12PCxe3MP+\ngGefBY9HOkgDas+7Gve/FmMaX0T6J//ssdTX9813oW32K0n7zFVag3PHbszJ8Wz435cpf+0TvHXN\nAJiTHaSfOJ3Jv7ma+MKBGSd2v6HBtuqDsz75Kc6yBhy5yft/jQ816GIu2O78PfjpexpsK6WUUjGp\nm8DVnhKqprunm9S9qUbel+32V3U1XHstABW7DQQNeQAY4h1kfPdRj+0nzdMmYpk+GdtF52A77fiB\nTVMPgu3O94cF10bSkuRg5DUnM+H+S7HnpA5JWpTaXxmNRhLy9f9IDYyYC7Y76+9vsFJKKaWGhiEU\n/HmIdIpmMoeC4T0Fxe3G2e640V7W6aGDtD7pHNwHAlBbC++8A0ALDoKt0m7bPHEs6SsW9tpRUeaa\nJXuXjoFiMJB27GQKbz6LEecfGd20DBMGLdlWSg2xmA+2gxpsK6WUUjHtJmpw8T6HMpbrmCgT+1Oy\n7XTK+3DnVb1Vy97XgKm6Gl55BS69tMPkFhKwnnIc9vPPIP66S3tYOcpC5+W0lr9hdth7X1Z1pcG2\nUmqIxXywrSXbSimlVIwKlUr/gsmU0anzrb4G28EgxMdLSbe363A8XfbX03b31G46vN7993eZ1WzP\nIKNuR4eeqmOatiPtvyFuV6+UUgAx/22twbZSSikVqySCSUnt5naiublvm9i+XZ594R62+9ixWn91\nKtWsJItqMtg94iAcDaXDI9AODLOIMdaSqwXbSqkhFtPBdhCtRq6UUkrFqnCbbaujm4pyFkvXad35\n5BN5jotrN7GX3sj3Ntg+9lg4/3waL72RcnLxYybuqd+SXbYKY2/jgccgLdjeS1qNXCk1xGKuGnnn\n70Et2VZKKaVilQTFgWC7H++FC+W5qKh/mwoFvHGtNSRv3Qjr18OkSV2X29tI02ymPiGf1j//HaxW\nMnd8gTkna++2FSXB/o5JrpRSKqpiPm+0eC28+1K0U6GUUkqpnqQVJgMw7SdzoKxMJubn939DDz9M\nsrdaXv/2tx3neTySA19Xt1dpbP3b2xJoA9nlq4ZdoN2BFm3vhf6P0a6UUvtqWHxbL7gi2ilQSiml\nVGeGUMm2Jc7CdcE/ccRvfwDnniszX3yx7xtyOKSK+N13R6aNHBl5HQhEeiqfPLnf6Wx99wPqL7wR\ngMzyrzGmDdMxdLVke5/o0F9KqaEWk8F2+KtQf1KUUkqpGNZd8Pfaa7BjR/9KtnNyoKkJ3G78tOul\nPCwhIdJe+5VX+pVE38Zi6s+9GoDU914e3iXaYbFesh3jyVNKqaESc222e/LFB5CVB4XdNN9SSiml\nVBS1LzE0GmHUqP6tv3Vr20tnYjaJTeX4738Qk8cD06dDa6vsw+/f8/5D/JXV1J5+Cb5VawFI++Qf\n2I4+vH/pijHaZnsfBNHeyJVSQy7mgu2efkd+dLI8f6W/M0oppVRs2ZfquSNHSjvvUCDtbAqQCJh8\nHnjwQSnVBigp6ddmq2edRqCkHICk538z7APt9oyxXrIdq7QauVJqiMX0t7XG1UoppVQM29cf6oUL\nYc0aKC6WquTLlmE74YiOyzQ3w4wZkJvb5826P13RFmjHXXsx8VddtI8JjRFasq2UUsNKzJVsK6WU\nUuoAccYZ8pySAuUSHKc8kQbT/tVxuffe69dm6y+8AUxGMrd/iTl/xECkVO2FYKyN36ol20qpIRbT\nwXYZF1GFmTTmkca8aCdHKaWUUt0ZhCAmgAFjwWg45ph+lWq3/nMRgdIK7Od/b78JtCsXr2L1lY/h\nrqiPdlKGNw22lVJDLKaD7RG8RgpJ0U6GUkoppYZKlvQW7hpRhGPblj6vZgx4KDfkyRurheRnfz0Y\nqRtSrSXV/PfQ23BXRMYWzz53/2l7rpRS+7uYDraVUkopFcsGoQ1xVhblhjys0w/H0Y/V4t21ba/N\n0yZhTEke+LQNsPJ/fkbDV1uY+MBlHaYHAgFWX/U4pS8vhWAQc7KDI//7EElTC6KSzv2FjrOtlBpq\nMR1sd/cT7vWAxTrkSVFKKaVUT2Kgd+w0d0Xb65SXn4hiSvqmcvEqVn7/VwC07qpi1PWnS4/sJhNf\nnfsA3upG4gqzmfLEdWSdMUt7IB8IGmsrpYZY7AXb7b4I64HUTrM//Aecup90KqqUUkqpHvSz5+24\ngBPjqDxS33kRy6Rxg5SofRfw+Vh99RNSah1S+vLSDu8xGhm3YB4T7tH+agaO9uSulBp6wy6b9Of6\nu6OUUkrFhHA8fO3KpZx99tm8+uqrUUuLgQCm0flYp0+OWhr2pHljKe8nXtgWWKceObFLaaujKIc5\nu54b3oF2rJbCazVypdQQi72SbaWUUkoNC4ZQaeEzhxxP4dt/GdiN97Nk20AQg902sGkYQBXvrmDV\n3P8j4PaSfvxUDl9yf1vV8J0vfEDOOYdhTdNOYQeVBttKqSEW88F2AHDY4dbH4cHroXBStFOklFJK\nqQ6iXJJpa63HhB9TVkZU09FZ+T8/w11RhyUlnq8v/g0A4++9mPF3d2wPN+qqOdFI3oFFa5ErpaIg\npoPtIOAHJrjg+9fB+3+F7Py921ZVGWSMkEzNmt3ynJY1kKlVSiml1IDZQ8m2d/1mDGYT5nFF5NSu\nw4uFpMfuHaLE7Vn5m8tZecFDkQkmI0cs+xXpRx1ApQb9rJ0w6LRkWyk1xGKuUU1334PtcwQWvdLz\nul9/AocY5LFtfWT6OWPg9Dw41Aif/wdOzYFTsqVn81i2fjkseRHONkQe7/8p2qlSSimlOopGCFM9\n+Xiqxh9DuWkkCYEmdttGYcxIi0JKuir/52es/MFDHaZNf+rGAyvQjkE69JdSaqjFdMk2dKz1s2qZ\nPM+2wotfwMSDOi775F2R1z+YDJ+2gs0OpVsj0286JfL6CBt86R/a2m91FbDxC2ipg2MvlmHMfF74\nfh+HM/vjDfIIm3U63PZX2Pkd/Oxomfbzt+Dwswc+7UoppVS3DIPwQ9pLqaiveHvkTSAAQJ0jb+DT\nsBeKH3uL9bc+h9Fq5qgvf8OOJxeRccJ0cuceHe2kKaWUGmIxGWyHf15doWdzp/5OfF649GD4qt3v\ncHMjrPq443L/XQgLX+p9XydnwcJdYI/blxR3FQgAQVj+Iow6GEYdBDvWws3TIss8fhWcdQu883jv\n25p7F2xfLfcdK97tOG/lIri40/hoD5wTeV0wHbavgd+sgHGH7MsRKaWUUp2Ef4cHusDQQK/BdtXY\nozq835R9FHijW2oZ8Hj4+KCf0LxuF+bUBI5Z+VviC3OY/uT8qKZLtWPUkm2l1NAaFsF2MND78of0\n8N15xwWR1z9/Gh64LvL+nhfg3qugoQaOdsAbG6BgQv/T2lAB3lb4+61w8ZPg90DaKLjRJPMDwNeh\nZRNDj/baB9qvN8FHL8O7T0iAffRcMFs6Lu9sgsYqyC6E5jq460TYtlrmHXY2nH8H3NHuHmT7Gnm+\n7VC47gk4+YeyTZ8HbA549ifw9mOyzH3/gZnaR4tSSqmo6zkoan33gw7v4666CO8HdeB19bDG4PI1\nO1k6+SZcu6oBSJg6iqNX/BazvY9V1tTQ0WrkSqkhFpPBdji2DgJFRDK3H/kX/PRceZ09Up7f/XPX\n9b/wweGdjsxkhiU14HFFOkq796rI/AsmwuduqCqG7AkwbwZMmgULXuw+jVXFsP4DeOUGqAG2A8/9\nC3KB8dPBB9QBO9ut04RkIEwJvW8JPXLz4YEvIC4BzrhRHmF+H7iaIC4ZnLWQkAGOUMSemAaPf9M1\nbW+Hztfyf8Br98KZP4bf/xCe/rE8enL3yfL8Sg04ksFk6nlZpZRSalCFfvwDgQDV448m/rYb8H27\nHueTUmUtu24dTQ88TuL/3QUFP4xKEp07q/h09k9xl9cBMPKak5nx7M1RSYtSSqnYE5PBtrfd6wYg\n2yevx0yJTM8rBI8bFlzZcd1fPNd9kHj6JdI+ur0Pa+HZX8JfH5X319mgFdgcml+8Fhrr4OO34YhT\n4bPFkJoGP7oPFt0U2c72dtssA8rW9H5s4fj4hHNg51sQKIH/zQNHKoycKSX5P/kQdqyAh2Z3XD+j\nCB4o7nn77R35fXmAZDY8fmX3yz27XTpee+NX8v6S9I7zzVZ4fhekZEkV/s6l7fvipf+Foplw1A80\nw1kppYatQfwCdz7xHP7iHTTOv7NtmjEnE2NKMsm/vnvQ9tubluJyPhp7fdv75EPGcviiBVgzdJzs\nWBNwuii95ykADFqKoJQaYjEXbHf+uW7fYsvRrg52XRW8/ULX9c+5Wp4vvx1eelherwhE7gM+ewle\nvAJ+vhKaq+HW34LHCW/8CbqLkT9+O7Te4tB+a+H+UKA9DWg0SCKTUiUw784I4JUqKNsJl86KTP/o\nLXm2ANMBZx1s/Eim3WiSYc/8QDiPIAhUb4UrDWACfrECUnLlAVBXCvZEiOvmt/6kK+TRk8sflMc7\nT8Azt3Sc5/PA5dk9rwuQkg0/eQnuOVXeH3k+/PRVKZjY8S28ukDam1vj4PhL4frfwYt3tKtGfyHk\njoOyzVA4A/73X5BdILN2rgN3C4yZJdt7X34zOWN+7AfoXhcYTWA0S1r9XqmtYLaBuwkwdP28gkHw\nuaGpEla/DcXLIT5NpuVMghNvBtMAZngopdQ+G4zv4lDJdsujz3SZlb78rUHYYd8EfD4+P1F6ZDVY\nzYxfMI9xd/4gaulRHQUCAVo++5ad192Pe1sZQZcHgkGMiQ5G3HvdnjeglFIDKOaC7c7aB9vx7YLt\nbevB6468X1Tacb2bH4Kph0NCsrSp9rqhoVwC7SBw3ywJWAESMrvuNx2pHt6bb9sl8FdvwoITJXB+\nciPsroAFx8m8P26ApAx5PPcp7NgIR39Phh8DKe1eCSQB9tDzlj3sG+B/DgU3kLOd4UYAACAASURB\nVAnEAeGRzEzAQ1ul6vk3/4JJc6QdeWceD1gsHQPWs34sj7DNX0kA/JuLe09L/e5IoA2w/M3ue1j3\ntMK/n5FHWM4YqCiW/YC0Qb+2sPf9Afzppq7TTGY46FS45TlIzIz0NB8MwneLIW0kWOMhffTeB+qB\ngATK21fA+ONln2HBoGTevP8QvHd/37eZmCUBtLcVWmp7X/aN2+S58HDImw4ZhZLJkpAhj2AQ6kvB\nliAdBHuckJoHVoecf79HMgEMRqm1EJcs6xR/Cs56CfKbq2V+YpYsY7IAocwCeyJMPkW2YbZC9nip\nlWEKXUsBP7ibZbtKqf1b22/0IOZ8Bkor2l5bTzyK1IUvYbTbe0nM4Al4PHw861Zad1ZR8OOzmPr4\ntYO/0+FoKId5aadswZ8ov+/ZSPtDA8RNH8+IX/yQ1PNPjEqalFIHtpgPttv/fHeuvlxXJc/3vgSZ\nuZ3WM8CJoSrU13e6B1gPbAW+F9p+U1XH+aOBNCAZqVZuQzo2u+dL+HYh/PVe+BgJajOR3/fvt/sO\nv/oieH9lpO10ezOOlAdIb+rNjXB8KChpDD0qu67WrZLQcykyYHorUAtUA88VRZY7FgniS4B13Wzn\n4Yfhvffg4ovhtNMgP1/OXzAoPZiPnQXHzeu6XsAPNTtgy3+hYhe8cBfc/CeYfCR89i68FKrxZ0+E\nq38jzQB+dZS0U/8OyZg4BDAUS+l/+7b67c+BKTStfT95ViKZC+35ffDVQrgsJzItFWkr39ppWTOQ\nkgEjxkDmKMjMgE0fQcWGbjbcjWDo4Uc6OM2aCNXrZZoxNM+A9EYfLpUuXQMjJktptTUOio6EVW9I\ngGu2Qd40KJwttRzMNph+Jhx/k5SOe1rhk5fhnV9CwAcbvoBNX8g+zHTtFNiFXLv1SHOMQChdZuTc\nW5C+BXxACpJhE58qQXnORGipkZoWRpOkDyBzjExfeJ+s52m336R0CHikEz8TkFkACSMgLR8yCmRe\nQwVs/ARa6iEtAxwjIMEh830eCdCTsqUmQP50SC+QoD7ol+HyggGpyWFMAEsiZGZFrtXwc2M1tHgh\nMxPM5kgMUF0N5eXQ3Aw7dsiyDQ2wcydkZUF6Oni9snxCAqSkQF4eZGTI/MESDEomi9UxePtQ+84X\nas5kjvlfzWgZ+GA7GAzi21kKfn/btPQlfxvw/fTVxntfZfOCVwHIu+Q4DbRjUP3bHwNBUi86hRF3\nXUPclDHRTpJS6gAX87cNbYFDc6SksHCSlGx/+YHcGJ9xaf+2GR52eyFwAvAZEpgcMhM830hJ8+52\n+/cgpdxf/hsSRkqgDRJkRfLbJejwAF99DfmhTN1FX8HkGfDru+Gwo+Hy78G0g+G+xyWIOefIrulL\nRoKjK+bD9bfBTReBsQ7cCbAm1OC7IbRceR+O9+M9zL/9dnleurT35bKAFCNcdChUfBGZHg4qbcAl\n18PxodefIsE/TfD3HmpufYOU5HuM4ArA3TfBx7+HjUSCwWQkEyIc1AUAn0Eyzg/3S6DenhPYBTQT\nCSa7u9B9SABWXQ2EjsdCxz4DOjOG9m80QKB9ZkoQKtfLSz8S4KYBo6bCWhfMPA7yp0L2EbDlK6i1\nQUYuvP0qjJoiAZfTDdtrwdgADVVyfWwqgU8WS6/y9bsjx+8Ppd8aem4xgM8O9V7wWsFtgNIWMBvA\n18fSnmQbFNrBEoDELWBPh4Qp0NgMlUmyz7pQU4nMTKitggQksHYD1ho5z15CmVTbwb5dzpk3tJwz\ndG4agZadENwp6yQaIMkI8Ubw+MAZlG0YQw97aB/O0PE7iXxeCeFzYpDrsCXYKWPGKMu1dDOqgQFI\nskGTp9Pn2UmSA9ITITsJUhyQnQhGL5TUQZkTUhIg3QEzJkBdI2AGuwPMDvAawdAiowiYXRCfDNnp\nULMdvvwcygLgDPWF4PGHrnMD+I1y0vyhc5AYB45QWvOyISsHWj2weRuYjJAWB444sJihqhkMJjCb\nwO2BBidYjGA3Sw0hezxsK5X+LRwOyB8JNrv0a2ENPUpKZL7fD2lp8pknJ8OmTRJwjsyFhDjwuSBr\nBKTmQFwcTJoAGaHMiZoKqN0OPqcEqxYzmOIh6Ia4eMifIJlO4doUfi/EpYSek4GgZEIEA9DaKM0p\nUvI61iYJ87oko8yeJBlVKbmSgRMMwo5tUFUq15DbC60uSEqSY6uuhqoqKC2FmmqorYXGJjl2l0vO\nxcaNklEDkgnj88k5sdkgMVGWq6+PBOR1dbJfk0n2k5Ul58zjke/Z9PSu6R+2wv83gzSkUtNt9wFg\nSEvBccPlg7KPvtj5wgdtgXbqUZM46C+3RS0tw0mwl+HbBmV/rW6McXaKXn1wSPerlFI9MQSH+ptw\nD9asgBmHNRIOOY8jic4jco07G157O/L+qz0cwbmhm3AHsLQPaZiMlAJnI4FgWAvQvtAziNz0G5Eg\nu6XTdjKQzABLp/SFb55dSLBUH3pvI1LiGOYNbdeBBFX1fUj/ZWfDkrcj6WvolKb22w8iGQZm5Bz1\nFmjui3CQGpaSAJ7mSNDUGyuSNvcelgs1n+9WUhz88Eq46ArYsAlmzoTx42Hth9BUA7fcCOXNUNXD\n+iClv9OJDN827kioLYfcmfD4u+Dy9TosLCYiwaEHySxIC0jNhPCx2Q1yzSSaJDByeaE1CL4AYIbW\nPXxA8XZwWCHVISX1DguMHglHz4HUFGhtgcZa2LwKSrZAMBSMVLbAriaocUODKxJ4mpB76CQjGANg\nt4EtEXbVSnDWGLrozeZIoAGQnAgtrR2nETrm9DSYPgUKsyRNZWWwaSNUVENDKMJOcUCyJZSxYoSG\nFjAFISkAfifYg2AxQZMvMkQgcoqwmSDdLie7tj7S90ECkBIHGCDZIBM9LlkngPx/hWtR+ELruEKP\nxtCzs91z+PPKMIE7CM0B+az6K8kk//suv9QuCNeKsJulBoMRuSZ8BrlubAZo8kcyMJJCaW0NpduA\n/M+0j0ftoW36Q/8kPuQ7xRhaxx2aHzTII2CE9Hj5vIIGue4aW8ETkAwRQwAaA5GaDXv7I2Ig8n0U\nrr2SgGRUWYl8Z/iJZHxaDOAzSyZSMACuIFgN0rzDFjomS+h8VIWOb0/pM4b2a0E+A6sRLKGMi1Yn\npBggM0maaXgtkrnR2Cy1KJrdEGeHpATJQDBaINEOZrtsuLEJ6hvB7ZLEvPIhFE3eyxMWg9bHjWGy\nayvbzr+GgjeeHbDtlptGYjn6UPybtxFscZLTsLHX5T8YdTX+Zhen1v51wNLQ3r+zLsNT1cCsf9zJ\niPOOGJR97E9c20r5rugcsn56KSN//T9Dsk9ffSPf5p0BZhMHNSwbkn0qpdSexHTJducxqcPWvd3D\nDKQDs6np8Lu/QONH8PPn+r/fcFXrRuDPf4YpJ8G1J0qQFjbTDO/5uls7ohq63OWFS4q607mac3u9\nBaXzL4ZHX5ASmDCvB958Ch65BY46EzaWwsuLYNEbcGe7ts4GZLiyzgKhednIUGVNQC+nvU1cu+Po\nblzxNs1ys5+CBC/h6s0V3SzaXXXxiRNh1sHwSrv7qvanOpx5QSg9ja3w2yflsbfq6VRLYHnoeVvX\nZW1EghCHUYIUD5HgDYBAJPPEBuSPgPpWCawrG+mQ82E0wrFHwebNcNZZUvX5+ONhwwYpbfzeGeBs\ngLUr4cP3JPjNGwXJqVBfC8/8DpJT4JRzwBEPI2fBMRfCMXMgJRVqqqRU0x4HDfXw4fvQ2gzffCnr\n542SeQYDpKbDuMnS3v/1F6F4owxfZ3NI6azfB4VjYGQh1NZB7kiYcTC0toLLCWMnSK/+fdHYAFYb\ntG+eGQzKw2iUGgClGyPTK7dD1S6oKZFA6KBTZHp8MlSXyPyAHyx2aW+eFqoWYTJLJ3/uVulbobUR\nSjdBYyWMO1QCr6qd4GqBhkpoaZJ+IGp2QvUumWaxgyEJ4hyQVwQTjgGrCUwBCdicTeDIgpZm2LgW\nqkogJ0UyRTJHwdTjZTg/rxt2rIXmOnlvNIWqxtfL+W9tgpYGyBotpcKpORCfIs81pXI+EtKkNtDu\nbZKulnop5S3bDFU75BzY4+W4TWZoqoXEFAlevR6or4JAA90yGmSdxFRITJdSZJdLMi6aWiCQLAEm\nQSnpNpoBs1wvFodkEHj94A9KKbI3FElbrXINNwNlLVJqbzNLEG0xScaDBWjxgdkH5qB8RyUjgXc4\nGA8i27AAhaH/LTsSkFtMEkg7bGC0ybmJM0KcTUZaCJeEe0O5D1YbJORKJlXQD/5aqS0SDEKWFez5\nMmRjwCd9IbQ2QGuNHGvQJ+c5xwIFCRCXAanZkhG1PzF0eTGAGw4ECVRUYTn8oAHeeP/seHoxnqoG\n8i47XgPtGNXw3qcUn/dTgh4vCcfP2vMKSik1RGKuZPvbr2D6oVKy7aCBI0miqNMyAeBrIiUhX7c7\ngqkZULenns2ApSvg+EPl9diJ8OgTcNcpst3uuvXwEAqe92DFCjjoIAkGt/Sll7N+euIJuHmAh/AM\nBODUnEgb+LBw9fCeBIGr74TkPHj1z/DNCgnkkloipczhdsLhkulaJPh0BODcy+Gnj8v+N6+Bg4+F\nL5dAdVnHId3iMqRt+9Fz4PLbYNbxEnS074+nqVHef7gIPl8Gf/8zXH4jvPh7CQT8yA24FykxC9Kx\nJoKDSJXono45gNQSMFkkoGpskGVNSKbC3MulHfKP7ohUo+1wvkLX6V23S5vsylL46P1IG+/uzi+9\npAcimTd2uxyn2QzHniKfw+4yaG6S81IwRtojf/heqAQwFCCBBAbeHkrM80dDdq4E3PW1EpC0NEdK\nrO1xcOo5ULVbPqOWZinxLt0hQXtPsnJgzEQoGi+v0zKgthoqSmFzqDr+1k0yzWqF/AIJ2k0mCfat\nNphxCBRNkEyElmbZTnOTpNNgkOrSJpNkIjQ1wOqvJEjOGgGN9ZHMg8oK2XZGFqRnyrXk88p2UtMl\nwyIzBwrHRqoG91UwKJ+PybTn/qOqK+U87i6Ta8tul3Ub66VKekuTnHe/T86TN5TG3WUSzNdWyzbS\nM2FUUaRtsd8vQbQpFPB6PPIZBYOQPUL21doi7xOS5LPwemD3Lmiok+ro5SXS1t3rAatdjsdml7R5\nvXIufV65tgwGiE+Q/xFHPBSMlc+ioQ52bZPS35xcSY/NJvvNzIER+fI6JU32F/4eT0ySa9BgkM8y\nPkGaMNRVgqtJMjRSUmXb6VnQUCOft8EEyQmQlAYEob5arlGfHxpq5XP3OiXtfr9kyAWMUmrtD0gV\nfE8rmPyQkS37aWiC8gpodUOiWTKXDD5Z3+eRUu2UNNmPsxW8wVDTCl+oQ0rgr6th3PS+X0OxbkPc\nGCa5trLtgmsp+PvTA7bdcvNIDMlJBGvrSfrjr4i/sfcq5INVsu2pbeQ/OVdgMBk5peYVzI5uOmZT\nXQxlyXbr+m2sm/wDMBoYcd+N5P786kHdn1JK9UdMB9twOkmYOYR5jKVjD10rkcDHCXxdDlu/gJQc\nOGl21222ZwM+2Qw3jpP1i6bAC2vh1VelgzCAeafIzf2zT0ug46djVez2SkshO7vnG/CtW2HBAvjD\nH+Df/4YLLoBFi6TN3j/+AffeKzfF4Y47q6vhssvg/vthVpQzZ1tboGw7XDg1uunoTV6hBACVpTB+\nprTnP+sqOHxO1+AmGITlSyWIK94I77wOJ50JRxwHJ5wOv/8VvPB7SEqW4CAYkOAhv0BKIu94AOZd\n07GT1e56dN8XdbWwoxjWrYYNa2HV5xLQnPQ9CW4++Y/c2OeNkmBlyky5fpoa4bRzYVShBEs98fsl\nzXFxsG0LrPhUgqC0UC/mrU45vkOPkmndZRr4fFCyQwLcovGyrc6CQUmTxy37KdkeKRWvKIPvvoZN\n62DLBgkYW50SXBaMlaArLh6KxsHoMRJAluyQofMMBqivk8Bu3equ1dT3JCFRAnKrVT5bn0+CuabG\nvm/DbJbgMD4hktlRVyPngyAkpch8nxd2bpNjM5lgZIEEjYXjZL9mswRmbhesWQkrP+t7GrJyJMMh\nI0sCwfgE2W9yigTPJdvl+NoyB0K5Xy3NoQDREGqvvFs+75Q0ub6aGuV6CAYlgLWHqt2PnSj/Fza7\nfKbBILhaZZ9mi/wPGkIl3gaDBJtul5zrbZvlOSFRaju4WiUwtlhlW34/VJZL7YpwJ3dxDtmf2SKB\nc2NPX8D7wB4n6Q5nhozIl2ve75dMAmcoA6Jsp1xzIP/ro4rkvBdvlM+xuTFy7CDHZzbLuUnLkP/f\nxFAnmJXlsHwrpO1HbbbDwfacEaOIO2QG8+bNY968bnrU7Kdy8yj5MAwGsp1buu99vJ3BCLYrF6/i\ny9MWAHDIO78g58xDB2zb+7vBDrYDThffTbsQz9bIUDT5j91G9i37fu0ppdRAivFguwFIoru+tWqJ\n1OI94TjYvaz7nrYBvq2Gyw6B5u2yVaNJqvcBPLNNqmN2HqXitNPg/fe7buv735cgOTERKis7Vm/d\n3zmboaUR4pOkJNpkhkd+DGu/6LjcQ3+HORd0nBYIyDku2QrffAJz5sK7L8JD82X+mKlQvFaC3Ixc\nePA1GD8j1FY1VGL29Sfwx5/Dd1/2Lb2X/RR+eHfHIeNUbHK5+v+/1NQoJbw1VRLMlJdIAJWTKxkK\nrc5Qx3PNEryNmShBTrgZhy/Uxt5ul8CqqVGCoXBmRVqGBHlVFVBeKpkgRqMEiDVVoVJSn5SYJqfI\ndWa1y3SvFzBI5oE5VFV713YJMrdvkfQYDBKkG4xSaj7nLBhdJAFpapoExo54CUpBAurw99RQjKrj\n90vptskMyelSkmwyQ2WJVMEPnwtPqIaEwSAZdIkp8v9aXS7fFa4WsMXJcQb8cjxmiyyXPVKeQYJU\ng0E+h9T0jhmY4cDX65XXScmREunqSqiplIwMV2ukkze/T4LkpsZQR3Dxct5t9lDv9g3QVC9paqyV\n0u6AX9KZM1LSaLZAdQXU7pZ9GACCUnqOQdKeliVV8htq5DhdTmmCkZAMGSOkRoI9TkrYXU44ee6g\njpI15DbEFTHJtY3tP7iW0X8byJJtCbZN44vI2vjJHpcf6GDbubOKj8ZcR9DnJ+f8IznkjZ8NyHYP\nFIMdbG+79BfUvrIIAHNmKo5DJlH4t4cwJ+iwDkqp2BLTbbbDWpG2wO21b/L5Uad+MIqAHCQY/3In\nxMeBabt0JBQAKapGquluLoFTz+26z+4C7djKlhh6jgR5QGT4shc/79u64eAgv0geABfcKI89CQ/5\ndvgcebTn88m45WOmwLqvJPB5/Xfw6Xvw8iPyCDv9EikBX7NcSrTCjj9XSvALJsLcm2DsVLlR9nnh\nnRdh1cfSq/IFN8KRp0Fjndycj5kqmQ5//j/JKBg/E1IzJcCoKpWAI7dQSmk/XwyjJ8Cp82DCQTLP\n45L2nj0NJeR2SelltG/MfT6o2AnJaRJc1IfaUzQ3QFUZpOfI5+tqleWaG+TYwkFU0RRIz5ZjbaiR\nY2+uh9QsSEqV4+tLoN3aIoFLQw1s+kb2U1kin92o8bKP9BwgEyq3StCbMxJSEyXgLtsC5VtDgVCK\nfIbffSnPCSmQnQ/1NbB9veynsRbWfyWf35ip0nxg904oKZYgz+uWY+mO2SL/K5+2yLCEZotUBU9K\ng3EZErSOnSbp2rYe7D747gOoyodv4yAtW9ZLTJFjstqg3CXnOTFFzntJsaRx1HhwJMo1V1spwWYg\nEOrBOxT8NtTKNVy8VoJQo1HalcfFyznw+eT6N5nk2Ct2QmmxrDfYktMkaG9tibxvaZJrw2yRzya3\nQP4nW1ugqU4C4sZamW8yyzVhtctnW10eCsw9oWYENvksPO7QGPCtPfeb0ROTSbZvD2WSJKfLs7NZ\nzqvXI5ktrS2yv4Qkmdfa0nVbEw+GUeP2+bTFnoH+ogp9SPa5Zw3sdvvAU93Ih4U/hECQw/59L1kn\nR7fNuBKBQIDK3/yFykf/irdcfohy7rqGvF/24UZCKaWiZFiUbB8GzGy3TACpRr6707rhA5mN9Lqd\ntwBaP4F1S2T6cmQIrzOREgrD9+DthZH1v/gCCgsjY+oefjgcfTSsXw9vvaXjqw4nbhe8+BA8c2+0\nU9I3510nN+h1VbBqmQSQYXlFkDNKgv/xM4GgBA5N9RIExCXIjX5yhvRybLFKYJKaGQl+W1sk0EpM\nhV2bJegzmWX90m1Sgjl+BuzcJMG02SqBRX01lG2TEjmQ/bQ0Ddxx2+xyfJMOkSYAzmbZb2MtZOVL\n0LhrM3z7OWxe3TXDKzldSkd379y7wNBml6C/sVb2bTDI2OtZ+RLkTpgp9/ybvpGAKjNXMlAciRIA\np2TIcobQUHQNtXJOy3dIaWhqlnwGLqdkQISDr/DnkZQqQa7XI5k4tbulI7Xq8p4D+fbMFrkuwkym\nSPXxzseZlQ8jx8lrowl2bZF9FUyU87hzk6xni4MRo+VYp86WALWlST4XkwlGFEjA7vfJOTBbZHvB\noLx2NklAm1ck16k11PmYxy3zA/7I8ZZulXNltkBKugTRFTslg6CpXvbRVC+Bv7NZzrsjQZaLT4x0\nlpeYItsMZyIEAvLZOJvku8DVEimNj4uXDLGsvEjQnJwu/zOW0NAHFTvlGDwuyMyTDIBwDYPO2reJ\nDwQiJfKBQKQmkMclx2syS6bQ/qStZHvudYx+/U8Dtt1yQx4A2VXfYszYc4+KA1my/dGkG2nZUErh\nT85mym9/uM/bOxANZMl2IBBgx5ULqHvt3wS9PjCbSL/se4z6/e0YtQ29UirGDYvwsX3hwIJ1cEg3\nw6YEiPRk3QjEHSNtpQHC+eLhftO+I9SBdLtAG+Cww+S5rAwyMqR9nhqebHa4foE8wmorYet3odLF\nUAdFhtCNdVUZ7NwsJZn/+ZuUVh16Ihx3Dpx2sdyI/+ke+NezchN98LGyvfEzJPi69DbYsEpKKQ86\nRkoLCyZKaa/PIwFxUip8t0JKTltbYNlbsi1nE7z3UmhooJCcUTD5UAl0N6+G8u2y7OpPZX7RFAmm\nfV7pUK451KlXSy9tj5PTZX52fqitab0EL3EJUur+4ZuQMxrGTpflfF5Jx3HnSE2G8HnKHyMBmSNB\ntlldLqV+HpfMy8yTwKSuUoKojd/I9qrK5BwkJEtwVFcl2ywphvUr4f1XZPlgUNLVWCvHnJoJs0+F\nH8yX6YkpkqmQli2BXJjXI/so3SrBYkKy1GQIB5DOJgne4hIkAyIuXmpEmC2yn6b6SEl7tPm8ElxW\nlUlaA/527aEbJJDMHinnv2ybBLJpWTI9HGiH22UH/KGAeAiqnvfXzKOinYLuhWvf9IXJhPSSSMeq\n7+FaCCAZFfb9vnbrwObbW44+DN/aDX0KtAEMA/SPu/yYO2jZUErO+UdqoB0DPDsrWDvxfIKtbgw2\nK/lP/JTMH83FGItfaEop1Y2YK9leuxKmHdKxZHsKcBRQdATcsRzyuvlNrSMy5NRsoH3t5tlIR2pr\netjnmjUwZUps3oyqA4PLKY+UXjo3Awm07I5I1fr2PG7Y8q0EpPVVUhqZMUJKKL0eCbKHk/ZDfCml\nYtMGexGT3NvYPvdaRr8+cG22+2vJ6GvwNbXuU8l20/pdLJv8IwDOcL+B0dpDdQa1R+4d5awtOIus\nWy9h5G9+0u/1nd9sZOvcO3Fv3glA5k0/IP/Xt+yxozyllIo1MVeyHeym6mS8A3DCmCO7zpsIbKDj\nGNWdmxH31qx48WKYNq2/qVRqYPW15Cshued5VhtMPiT0ZsKAJCuqOg/vppSKZcM3V6xy8SoyTp7J\nF6feAwYDx337Ow20o8hTWcumE2/AX9cERiNZ/zNvrwJ2pZSKBTEXbHfH5YRTbofT7pD3RxwPny2V\n1/HAacDzfdzW++9LT+MgVSy11EwppZQa/vamml7p6x/z9UWRnjQLbj6TxCmjBi5Rqs8aFn/GltN/\n3NYWJu2S0yn8yy+jnCqllNo3wyLYXgNMPBPi0+Q7OBxoj0E6Fq/vYb2zABfwn9D7KVPg1FO1V3Gl\nlFJqYIR+UIdpxvW3Nz4VeWMyMvkxbac91AI+H5tP/hHNS1e2TRu/7GkSjz04iqlSSqmBMSyCbYDS\njbC5RoZhCus88uYMYHXo9ZTQ8/t+Ca6bmiAlZfDTqZRSSh1ogsOwzUfxY2/hq2tue3/sN49px1tD\nyFW8i9Kf/Z7GRcsJtLQSP3sapuQERj9/N9bczGgnTymlBkTMBdvd/V6nAW4vnHeevB9B12G/AD4o\nhWuugrf/DXnAZQ9GqolroK2UUkoNjuEWawc8Hjb+78uY4u0U/eRsXBV1JE0tiHayDgieylq2nvdT\nWpaHuq01Gsl75BZybrssuglTSqlBEHPBdneSgcq6yHs3MtRXZxm58M9F0NiowbVSSik1VIZbsP3R\n+BsJtHqY/syPGPXDU6OdnAOCr7aB7VffR8NbywCwjMgg4/rvk3PH5drLuFJqvzUs6kttA74rjryv\n7WaZM8+UZ6NRA22llFJqb1x11VUYjcYOjzPOOCPayRpQ2/+0iNYdVQAaaA8BX20D3027kNXpJ9Hw\n1jJMaUmMW/Ik08veJ/ee6zTQVkrt14ZFyTbAo710N+7xgKWbcYeVUkop1T+nn346L774IsFQb6I2\nm23PKxlNg5yqgRHweFj3k+cAOGlXX8cxUXur+rm3qPztK/LGYGDsu4+RfMZR0U2UUkoNoWETbPdk\n82YNtJVSSqmBYrPZyMzcPzuoWjbjFqk+/uzNxOVnRDs5+y2DRW4vAw3NYDIy9u1HNchWSh2QhkU1\n8p5ccAGMGRPtVCillFL7j6VLl5Kdnc3EiROZP38+tbXdNd4KGUZDaTat30XLhlJMDhujrjk52snZ\nr1lzM8n+f5cx/r/PMcv3pQbaSqkDVsyVbPenk5W//33w0qGUUkodaE4//XTOP/98CgsLKS4u5s47\n7+SMM87gs88+w9DtD3Qo2h4GPaStu1Wqjx+z6rEop+TAkP/wLdFOglJKEDQ6BQAAGh5JREFURV3M\nBdt9dckl0U6BUkopNXz99a9/5frrrwfAYDCwaNEi5s6d2zZ/ypQpTJs2jTFjxrB06VJOOOGEHrd1\nzYcLiTt7R4dp8+bNY968eYOT+H4K+HxUf7AaR1EOCRPyop0cpZRSB4iYDrYTAQPQ2M28v/xliBOj\nlFJK7UfOOeccZs+e3fY+L69rEFpYWEhGRgZbtmzpNdh+/qQzGfniE4OSzoFQ/Ot/EvT5Kbr9vGgn\nRSml1AEkpoPteKRR+QRgRZTTopRSSu1P4uPjKSoq6nWZkpISampqGDFiRPcLxEqb7W5qsXvqm3GX\n15E4aSTb/7AQo93CqGt1qC+llFJDJ6Y7SEsKPcd0IpVSSqn9QEtLC7fffjtffPEFO3bsYMmSJZx7\n7rmMHz+eU0/tPUgNxsIPdbBj5P+fzEtZNvlH+JqduMtqSZ09AaMxFhKqlFLqQBHTvzqO0LO33bR0\ntGM0pZRSaqCZTCbWrFnDOeecw4QJE7j22ms59NBD+fjjj7HscYzN2LqdaFizjaAvAMAHeVdDEAp+\nfFaUU6WUUupAE3vVyNtVBdvGRbgwUzRxHpkbpJMVCzLkl1JKKaUGjt1u5/333492MgbE2vlPtb32\nNToBGHHeEdFKjlJKqQNU7AXb7YzjNcpI4oRTYOuGaKdGKaWUUt0yxs7QXw1rtlH36foO08bfe3GU\nUqOUUupAFlv1vjoJt9mOT4xqMpRSSinVmxgaZ3vzL18HYMoT12KwmMg8ZSbj774oyqlSSil1IIrp\nku2RwAbgq+WRaf/+JlqpUUoppVR3YiXUbi2ppuINuWkovPksCm/WdtpKKaWiJ6ZLtg2AyQjLP4pM\ns9milhyllFJKxbANP/szADnnHxnllCillFIxHmwDmDulcOzE6KRDKaWUUj0wxcbtRMW/Pseamcwh\nb/ws2klRSimlYi/Y7tzsyxQrddOUUkop1UF6oXSqkliQEeWUgK/Bib/FTf5VJ0U7KUoppRQQg8F2\nZ62+yOvH/hy9dCillFKqI1uCFQCjyRTllADBIADjfzE3yglRSimlREwG2+Gf7Asfb/vtBOC4U6KS\nHKWUUkp1p/2PdBQFfP621+YERxRTopRSSkXEZLAdduSVHd9n5UQlGUoppZTqjTG6txPemiYAss85\nPKrpUEoppdqL6WBbKaWUUmpPAi4vANP+eEOUU6KUUkpFxFyw3bmDNKWUUkrFqBipRh5mz02PdhKU\nUkqpNuZoJ0AppZRSw9tFzz+Pedky5s2bx7x584Z8/1P/eAPuyoYh369SSinVm5gPtpdtgON0bG2l\nlFIqZr12zTUk3Xpr1PZfcOMZUdu3Ukop1ZOYq0be2dgJ0U6BUkoppXoVY9XJlVJKqVgQ8yXbAEef\nBLOPjXYqlFJKKaWUUkqpvom5YDsQ6Drt9Q+GPh1KKaWU6iPt3VQppZTqYq+qkf/hD3+gsLCQuLg4\nZs+ezYoVK3pc9tlnn+XYY48lLS2NtLQ0Tj755F6XV0oppZRSSimlhrt+B9uvv/46t912G/feey9f\nf/01M2bM4NRTT6W6urrb5ZctW8bFF1/M0qVL+fzzzxk5ciSnnHIK5eXl+5x4pZRSSkWRttVWSiml\netTvYPvRRx/l+uuv5/LLL2fixIk89dRTOBwOnn/++W6Xf/nll7nhhhuYPn0648eP59lnnyUQCLBk\nyZJ9TrxSSimlYoAx5vtbVUoppYZcv34dvV4vK1eu5KSTTmqbZjAYmDNnDp999lmfttHS0oLX6yUt\nLa1/KVVKKaWUUkoppYaJfgXb1dXV+P1+srOzO0zPzs6moqKiT9u44447yMvLY86cOT0u4+9PopRS\nSimllFJKqRgzIL2RB4NBDH3oifShhx7ib3/7G8uWLcNqte5x+XHjxmEwGMjLyyMvLw+AefPmMW/e\nvH1Os1JKKaX2kbbZVkoppXrUr2A7IyMDk8nE7t27O0yvrKzsUtrd2SOPPMLDDz/MkiVLmDJlSp/2\nt3nzZpKSkvqTRKWUUkoNNR36SymllOqiX9XILRYLs2bN6tC5WTAYZMmSJRx55JE9rvfrX/+aBx54\ngMWLF3PQQQftfWqVUkoppZRSSqlhoN/VyG+99VauuOIKZs2axWGHHcajjz6K0+nkyiuvBODyyy8n\nPz+fBx98EICHH36Yu+++m1dffZVRo0a1lYonJCQQHx8/cEeilFJKqaGl1ciVUkqpHvU72J47dy7V\n1dXcfffd7N69m5kzZ7J48WIyMzMBKCkpwWyObPbJJ5/E6/VywQUXdNjOPffcw913372PyVdKKaVU\n1OnQX0oppVQXe9VB2vz585k/f3638z788MMO77dt27Y3u1BKKaWUUkoppYYtzYpWSimllFJKKaUG\nmAbbSimllNo7oTbbFz39NGeffTavvvpqlBOklFJKxY4BGWdbKaWUUgeu166/nqSbbop2MpRSSqmY\noiXbSimllFJKKaXUANNgWymllFJ7R4f+Ukr9//buPDaqev3j+HOGdqAtnVK2lmLZLlKLAkIVKRCu\nCrKIlHhBLQJVIzeINArkWiEK5BcXMFwloKLGLXiVDpElYFgsaayIVPa2hFrAsBSX0pqUGUUtlj73\nj2vPr9NhKHSms/X9SprQc75nzvOB0jPPfM8CwCOabQAA4B0e/QUAgBuOjgAAAAAA+BjNNgAAAAAA\nPkazDQAAmodrtgEA8IhmGwAAeIdrtgEAcMPREQAANI9hBLoCAACCFs02AABoHk4jBwDAI5ptAADg\nHWa4AQBwQ7MNAAAAAICP0WwDAAAAAOBjNNsAAAAAAPhY0DXbDe+1kpmZKRkZGZKbmxu4ggAAwNXx\n6C8AANxEBLqAq7Hb7WKz2QJdBgAAAAAA14WPogEAQPP8dTpa5htvcCYaAACNBPXMNgAACH727Gyx\n/fOfgS4DAICgEnQz21oX6AoAAAAAAPBO0DXbAAAAAACEOpptAADQPA0fIQIAAFzQbAMAAAAA4GM0\n2wAAAAAA+BjNNgAAaB5OIwcAwCOabQAA4B0LbycAAGgs6I6OfEgOAAAAAAh1QddsAwAAAAAQ6mi2\nAQBA83A6GgAAHtFsAwAA73DNNgAAbjg6AgAAAADgYzTbAAAAAAD4GM02AABoHq7ZBgDAI5ptAADg\nHcMIdAUAAAQdmm0AAOCVzFWrJCMjQ3JzcwNdCgAAQSMi0AUAAIDQZn/6abE9+migywAAIKgwsw0A\nALzDo78AAHATdEfHhvdayczM5LQ0AAAAAEDICerTyO12u9hstkCXAQAAAADAdQm6mW0AABAiePQX\nAAAe0WwDAADv8OgvAADc0GwDAAAAAOBjNNsAAKB5OI0cAACPaLYBAIB3ePQXAABuODoCAAAAAOBj\nNNsAAKB5uDEaAAAe0WwDANAKbN68WcaPHy9dunQRi8UiJSUlbmNqampk7ty50rlzZ4mNjZWpU6dK\nZWWl5xflmm0AADyi2QYAoBW4ePGijBw5Ul555RUxPMxIz5s3T7Zt2yYbN26U3bt3y48//ihTpkxp\n+sW5ZhsAADcRgS6gMT4kBwDA92bMmCEiImfPnhW9wsHW6XTKBx98IHa7Xf7+97+LiMiHH34oqamp\nsn//fhk6dKhf6wUAINTxUTQAAJBDhw5JbW2tjB492lyWkpIiPXr0kMLCwitvxCfkAAB4FHTNdh3H\nbQAA/K6iokKsVqvYbDaX5QkJCVJRURGgqgAACF1Bdxo5AADwzrp162T27NkiImIYhuzYsUNGjBjR\nrNdSVY/XeNfL/Pe/JSI312XZtGnTZNq0ac3aJwAA4YBmGwCAMDN58mQZNmyY+X337t2b3CYxMVEu\nXbokTqfTZXa7srJSEhISrrqt/V//EhuNNQAALoLuNHIAAOCdmJgY6dOnj/nVtm1bl/VXmqlOS0uT\niIgIyc/PN5edOHFCysvLJT09vcVrBgAg3DCzDQBAK1BdXS3l5eXyww8/iKpKWVmZqKokJiZKQkKC\n2Gw2efzxx2XBggUSHx8vsbGx8tRTT8mIESOavhN5E6eZAwDQGjGzDQBAK7B161YZPHiwTJo0SQzD\nkGnTpsmQIUPknXfeMcesXLlS7rvvPpk6darceeedkpSUJBs3bgxg1QAAhC5Dr/SwzQA6XCiSNtwp\nInHicDjc7ooKAACCg/Nvf5O4U6fEkZsrtszMQJcDAEBQYWYbAAB4x8LbCQAAGuPoCAAAAACAjwVd\ns93wnPbMzEzJyMiQ3EbP7gQAAAAAIJgF9d3I7XY712wDABCsguu2LwAABJWgm9kGAAAhhmu2AQBw\nw9ERAAAAAAAfo9kGAADNw2nkAAB4RLMNAAC8YxiBrgAAgKBDsw0AAAAAgI/RbAMAAAAA4GM02wAA\nAAAA+BjNNgAA8A7XbAMA4IZmGwAAeCVz2TLJyMiQ3NzcQJcCAEDQiAh0AW54iggAAKHhr0d/2Rct\nEts//hHgYgAACC7MbAMAAO9YeDsBAEBjHB0BAAAAAPAxmm0AAAAAAHyMZhsAADSPcqMVAAA8odkO\nsNZ059bWkrW15BRpPVlbS06R1pO1teT0G67ZBgDADUfHAGtNb/haS9bWklOk9WRtLTlFWk/W1pIT\nAAAEDs02AABoHk4jBwDAo7Brtr2ZrWjutoGaIQlEvYHIyr9py23n7baB2GdrydpacgZyW3/vM6xn\n0w0j0BUAABB0aLZ9sC2NWcvi37TltvN220Dss7VkbS05A7mtv/cZ1s02AABwE+HvHaqq/PLLLx7X\nV50XEXGKiIjT6bzu16+trW3Wdt5sG4h9erMt9QbnPr3ZlnpbdtvWsk9vtqVe320XGxsrBjPFAACE\nPEPVvxdcOZ1OiYuL8+cuAQAIGQ6HQ2w2W6DLuCbOrCyJ+89/xFFcLLaBAwNdDgAAQcXvzXZTM9uO\nCyIbPnDKgv9LlnPnzoXMGw4AAHwhlGa26z9AD6UPCAAA8Be/N9vXgoM3AADBj+M1AACehd0N0gAA\nAAAACDSabQAAAAAAfIxmGwAAeCUzM1MyMjJ4vBkAAA34/dFfAAAgvNjtdq7ZBgCgEWa2AQAAAADw\nMZptLy1btkyGDh0qNptNEhIS5P7775cTJ064jKmpqZG5c+dK586dJTY2VqZOnSqVlZUuY86dOycT\nJ06UmJgYSUxMlJycHKmrq3MZU1BQIGlpadKuXTvp16+frF27tsXzebJs2TKxWCyyYMECc1k45fzx\nxx9l5syZ0rlzZ4mOjpZBgwbJ4cOHXcYsWbJEkpKSJDo6Wu655x757rvvXNZXV1fL9OnTJS4uTuLj\n42XWrFly8eJFlzElJSUyatQoiYqKkp49e8qKFStaPFu9uro6Wbx4sfTp00eio6Olb9++8uKLL7qN\nC8WcX331lWRkZEj37t3FYrHI1q1bA5br008/ldTUVImKipJBgwbJjh07/JKztrZWnn32WRk4cKC0\nb99eunfvLo888oj89NNPIZezqayNzZ49WywWi6xevdpleahkBQAAYUKDkMPhUBFRh8MR6FKaNGHC\nBP3oo4+0tLRUS0pKdOLEidqzZ0/97bffzDFPPPGE9uzZUwsKCvTw4cOanp6uI0eONNdfvnxZb7nl\nFh07dqyWlJTozp07tUuXLvrcc8+ZY06fPq0xMTH6zDPPaFlZmb7xxhsaERGheXl5fs2rqrp//37t\n3bu33nrrrTp//nxzebjkrK6u1l69eunjjz+uBw8e1DNnzuiuXbv01KlT5pjly5drfHy8bt26VY8e\nPaqTJ0/WPn36aE1NjTlm/PjxOnjwYD1w4IB+/fXXeuONN+r06dPN9U6nUxMTEzUrK0tLS0t1/fr1\nGh0dre+++65fcr700kvapUsX3bFjh549e1Y3btyosbGx+vrrr4d8zh07dujixYt18+bNarFYdMuW\nLS7r/ZVr7969GhERoa+++qqWlZXpkiVL1Gq16rFjx1o8p8Ph0LFjx+qGDRv0xIkTum/fPr3jjjv0\n9ttvd3mNUMjZVNaGNm/erLfeeqvecMMNumrVqpDMGkpC6XgNAIC/BWWzXVdXpw6HQ+vq6gJdynWr\nqqpSwzD0q6++UtX/vRGxWq26adMmc0xZWZkahqH79u1TVdXt27drRESEVlVVmWPefvtt7dChg/75\n55+qqpqTk6MDBgxw2VdmZqZOmDChpSO5+OWXX7Rfv36an5+vd955p9lsh1POZ599VkeNGnXVMd26\nddPXXnvN/N7hcGi7du10/fr1qqpaWlqqhmHo4cOHzTE7d+7UNm3a6E8//aSqqmvWrNFOnTqZ2VVV\nFy5cqKmpqb6M49F9992ns2bNclk2ZcoUnTlzpvl9OOQ0DMOtMfNXroceekgnTZrksu9hw4bpnDlz\nfBfwL1fK2diBAwfUYrHouXPnVDU0c6p6zvr9999rcnKylpaWaq9evVya7W+//TYkswa7UD5eAwDQ\n0oLyNHLDMMRms4lhGIEu5bpduHBBDMOQjh07iojIoUOHpLa2VkaPHm2OSUlJkR49ekhhYaGIiHzz\nzTcyYMAA6dy5szlm3Lhx4nA45NixY+aYMWPGuOxr3Lhx5mv4y9y5c2XSpEly9913uyw/ePBg2OT8\n7LPP5LbbbpMHH3xQEhISZMiQIfLee++Z60+fPi0VFRUuWW02m9xxxx0uWePj42Xw4MHmmDFjxohh\nGLJv3z5zzKhRoyQi4v/vUzhu3Dg5fvy4OByOlo4pw4cPl/z8fDl58qSIiBQXF8vXX38t9957b1jl\nbMyfuQoLCwP+89xQ/e+nDh06iEh45VRVycrKkpycHElNTXVbX1hYGDZZg0koH68BAGhpQdlshypV\nlXnz5snIkSOlf//+IiJSUVEhVqvV7S6tCQkJUlFRYY5JSEhwW1+/7mpjnE6n1NTUtEiexux2uxQV\nFcmyZcvc1p0/fz5scp46dUreeustSUlJkby8PHniiSfkqaeeko8//tis0TCMK9bZMEfXrl1d1rdp\n00Y6dux4XX8fLWnhwoXy0EMPyU033SRWq1XS0tJk3rx5kpmZadYQDjkb82cuT2MCkbumpkYWLlwo\nDz/8sLRv396sL1xyLl++XKxWq2RnZ19xfThlBQAAoYFHf/nQk08+KaWlpbJnz54mx6rqNc0EXG2M\nqjY5xle+//57mTdvnuzatUsiIyOvebtQyynyvxuHDR06VF544QURERk0aJAcO3ZM3nrrLZkxY8ZV\n62yqxqbG+DPr+vXrZd26dWK326V///5SVFQkTz/9tCQlJcnMmTOvWmMo5bxWvsp1LWP8nbu2tlYe\neOABMQxD1qxZ0+T4UMt56NAhWb16tRw5cuS6tw21rAAAIHQws+0j2dnZsn37dikoKJCkpCRzeWJi\noly6dEmcTqfL+MrKSnN2JDExUc6fP++yvv77xMREj2MqKyvFZrOJ1Wr1eZ7GDh06JFVVVZKWliaR\nkZESGRkpX375paxatUqsVqskJCRITU1NyOcUEenWrZvbaaipqalSXl5u1qiqV6yzYdbGd2K/fPmy\nVFdXN5lVRNxmzlpCTk6OLFq0SB544AG5+eabZfr06TJ//nzzzIVwydlYS+dqOGvuaYw/c9c32ufO\nnZO8vDxzVru+vnDIuWfPHqmqqpLk5GTz99PZs2dlwYIF0qdPH7PGcMgKAABCB822D2RnZ8uWLVvk\niy++kB49erisS0tLk4iICMnPzzeXnThxQsrLy2X48OEiIpKeni5Hjx6Vn3/+2RyTl5cncXFxZtOX\nnp7u8hr1Y9LT01sqlosxY8bI0aNHpaioSIqLi6W4uFhuu+02mTFjhvnnyMjIkM8pIjJixAg5fvy4\ny7Ljx49Lz549RUSkd+/ekpiY6FKn0+mUffv2uWS9cOGCy0xbfn6+qKoMHTrUHLN79265fPmyOSYv\nL09SUlIkLi6uxfLV++2339xm4ywWi/kotnDJ2Zg/c13p53nXrl1++3mub7RPnTol+fn5Eh8f77I+\nXHJmZWVJSUmJ+bupuLhYkpKSJCcnRz7//HOzxnDICgAAQkiL34ItzM2ZM0c7dOigu3fv1oqKCvPr\n999/dxnTq1cv/eKLL/TgwYM6fPhwt0diDRw4UMePH6/FxcW6c+dO7dq1qz7//PPmmPpHYuXk5GhZ\nWZm++eabGhkZqbt27fJr3oYa3o1cNXxyHjhwQK1Wq7788sv63Xff6SeffKLt27fX3Nxcc8wrr7yi\nHTt21K1bt2pJSYlOnjxZ+/bt6/LoqAkTJmhaWpru379f9+zZo/369dMZM2aY6x0Oh3br1k2zsrL0\n2LFjarfbNSYmRt977z2/5Hz00Uc1OTlZt23bpmfOnNFNmzZply5ddNGiRSGf89dff9WioiI9cuSI\nGoahK1eu1KKiIi0vL/drrr1792pkZKT5mKilS5dq27ZtffaYqKvlrK2t1YyMDO3Ro4eWlJS4/H66\ndOlSSOVsKuuVNL4beShlBQAA4YFm20uGYajFYnH7Wrt2rTnmjz/+0OzsbO3UqZO2b99ep06dqufP\nn3d5nfLycp04caLGxMRo165dNScnRy9fvuwypqCgQIcMGaLt2rXTvn376kcffeSXjJ7cddddLs12\nOOXctm2bDhgwQKOiorR///76/vvvu41ZunSpduvWTaOionTs2LF68uRJl/XV1dU6ffp0tdls2qFD\nB501a5ZevHjRZUxJSYmOGjVKo6KiNDk5WVesWNGiuRr69ddfdf78+dqrVy+Njo7Wvn376pIlS1we\ne6QamjkLCgqu+H/zscce83uuDRs2aEpKirZr104HDBigO3fu9EvOM2fOuK2r//7LL78MqZxNZb2S\n3r17uzXboZIVAACEB0P1rzsVAQAAAAAAn+CabQAAAAAAfIxmGwAAAAAAH6PZBgAAAADAx2i2AQAA\nAADwMZptAAAAAAB8jGYbAAAAAAAfo9kGAAAAAMDHaLYBAAAAAPAxmm0AAAAAAHyMZhsAAAAAAB+j\n2QYAAAAAwMf+Cw6m4Gqb6Py9AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "Graphics Array of size 1 x 2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nToGenerate = 15000\n",
    "iterations = 5\n",
    "g = twoRunningMeansPlot(nToGenerate, iterations) # uses above function to make plot\n",
    "show(g,figsize=[10,5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We talked about the Cauchy in more detail in an earlier notebook.  If you cannot recall the detail and are interested, go back to that in your own time.  The message here is that although with the Bernoulli process, the sample means converge as the number of observations increases, with the Cauchy they do not. \n",
    "\n",
    "We talked about $\\mathbf{p}^{(n)}$, for $\\mathbf{p}$ our Markov Chain transition matrix, converging.  We talked about sample means converging (or not).   What do we actually mean by *converge*?  These ideas of convergence and limits are fundamental to data science:   we need to be able to justify that the way we are attacking a problem will give us the *right* answer.  (At its very simplest, how do we justify that, by generating lots of simulations, we can get to some good approximation for a probability or an integral or a sum?)  The advantages of an MLE as a point estimate in parametric estimation all come back to limits and convergence (remember how the likelihood function 'homed in' as the amount of data increased).  And, as we will see when we do non-parametric estimation, limits and convergence are also fundamental there.\n",
    "\n",
    "# Limits of a Sequence of Real Numbers\n",
    "\n",
    "A sequence of real numbers $x_1, x_2, x_3, \\ldots $ (which we can also write as $\\{ x_i\\}_{i=1}^\\infty$) is said to converge to a limit $a \\in \\mathbb{R}$,\n",
    "\n",
    "$$\\underset{i \\rightarrow \\infty}{\\lim} x_i = a$$\n",
    "\n",
    "if for every natural number $m \\in \\mathbb{N}$, a natural number $N_m \\in \\mathbb{N}$ exists such that for every $j \\geq N_m$, $\\left|x_j - a\\right| \\leq \\frac{1}{m}$\n",
    "\n",
    "What is this saying? $\\left|x_j - a\\right|$ is measuring the closeness of the $j$th value in the sequence to $a$.  If we pick bigger and bigger $m$, $\\frac{1}{m}$ will get smaller and smaller.  The definition of the limit is saying that if $a$ is the limit of the sequence then we can get the sequence to become as close as we want ('arbitrarily close') to $a$, and to stay that close, by going far enough into the sequence ('for every $j \\geq N_m$, $\\left|x_j - a\\right| \\leq \\frac{1}{m}$')\n",
    "\n",
    "($\\mathbb{N}$, the natural numbers, are just the 'counting numbers' $\\{1, 2, 3, \\ldots\\}$.)\n",
    "\n",
    " \n",
    "\n",
    "Take a trivial example, the sequence $\\{x_i\\}_{i=1}^\\infty = 17, 17, 17, \\ldots$\n",
    "\n",
    "Clearly, $\\underset{i \\rightarrow \\infty}{\\lim} x_i = 17$, but let's do this formally:\n",
    "\n",
    "For every $m \\in \\mathbb{N}$, take $N_m =1$, then\n",
    "\n",
    "$\\forall$ $j \\geq N_m=1, \\left|x_j -17\\right| = \\left|17 - 17\\right| = 0 \\leq \\frac{1}{m}$, as required.\n",
    "\n",
    "($\\forall$ is mathspeak for 'for all' or 'for every')\n",
    "\n",
    "\n",
    "\n",
    "What about $\\{x_i\\}_{i=1}^\\infty = \\displaystyle\\frac{1}{1}, \\frac{1}{2}, \\frac{1}{3}, \\ldots$, i.e., $x_i = \\frac{1}{i}$?\n",
    "\n",
    "$\\underset{i \\rightarrow \\infty}{\\lim} x_i = \\underset{i \\rightarrow \\infty}{\\lim}\\frac{1}{i} = 0$\n",
    "\n",
    "For every $m \\in \\mathbb{N}$, take $N_m = m$, then $\\forall$ $j \\geq m$, $\\left|x_j - 0\\right| \\leq \\left |\\frac{1}{m} - 0\\right| = \\frac{1}{m}$\n",
    "\n",
    "### YouTry\n",
    "\n",
    "Think about $\\{x_i\\}_{i=1}^\\infty = \\frac{1}{1^p}, \\frac{1}{2^p}, \\frac{1}{3^p}, \\ldots$ with $p > 0$. The limit$\\underset{i \\rightarrow \\infty}{\\lim} \\displaystyle\\frac{1}{i^p} = 0$, provided $p > 0$.\n",
    "\n",
    "You can draw the plot of this very easily using the Sage symbolic expressions we have already met (`f.subs(...)` allows us to substitute a particular value for one of the symbolic variables in the symbolic function `f`, in this case a value to use for $p$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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dCQAACBRCUzW1b09LEwAA4YTQVE2pqbQ0AQAQTghN1dS+vfTDD9Lp06YrAQAAgUBoqqbU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\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var('i, p')\n",
    "f = 1/(i^p)\n",
    "# make and show plot, note we can use f in the label\n",
    "plot(f.subs(p=1), (x, 0.1, 3), axes_labels=('i',f)).show(figsize=[6,3]) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "What about $\\{x_i\\}_{i=1}^\\infty = 1^{\\frac{1}{1}}, 2^{\\frac{1}{2}}, 3^{\\frac{1}{3}}, \\ldots$. The limit$\\underset{i \\rightarrow \\infty}{\\lim} i^{\\frac{1}{i}} = 1$.\n",
    "\n",
    "This one is not as easy to see intuitively, but again we can plot it with SageMath."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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JAADACwITAACAFwQmAAAALwhMAAAAXhCYAAAAvPj//69KN4ABDCMA\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "Graphics object consisting of 2 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var('i')\n",
    "f = i^(1/i)\n",
    "n=500\n",
    "p=plot(f.subs(p=1), (x, 0, n), axes_labels=('i',f)) # main plot\n",
    "p+=line([(0,1),(n,1)],linestyle=':') # add a dotted line at height 1\n",
    "p.show(figsize=[6,3]) # show the plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Finally, $\\{x_i\\}_{i=1}^\\infty = p^{\\frac{1}{1}}, p^{\\frac{1}{2}}, p^{\\frac{1}{3}}, \\ldots$, with $p > 0$. The limit$\\underset{i \\rightarrow \\infty}{\\lim} p^{\\frac{1}{i}} = 1$ provided $p > 0$.\n",
    "\n",
    "You can cut and paste (with suitable adaptations) to try to plot this one as well ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "(end of You Try)\n",
    "\n",
    "---\n",
    "\n",
    "*back to the real stuff ...*\n",
    "\n",
    "# Limits of Functions\n",
    "\n",
    "We say that a function $f(x): \\mathbb{R} \\rightarrow \\mathbb{R}$ has a limit $L \\in \\mathbb{R}$ as $x$ approaches $a$:\n",
    "\n",
    "$$\\underset{x \\rightarrow a}{\\lim} f(x) = L$$\n",
    "\n",
    "provided $f(x)$ is arbitrarily close to $L$ for all ($\\forall$) values of $x$ that are sufficiently close to but not equal to $a$.\n",
    "\n",
    "For example\n",
    "\n",
    "Consider the function $f(x) = (1+x)^{\\frac{1}{x}}$\n",
    "\n",
    "$\\underset{x \\rightarrow 0}{\\lim} f(x) = \\underset{x \\rightarrow 0}{\\lim} (1+x)^{\\frac{1}{x}} = e \\approx 2.71828\\cdots$\n",
    "\n",
    "even though $f(0) = (1+0)^{\\frac{1}{0}}$ is undefined!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "power::eval(): division by zero",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-55-9fea08a5e631>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;31m# x is defined as a symbolic variable by default by Sage so we do not need var('x')\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[0mf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# this will give you an error message\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/home/raazesh/all/software/sage/SageMath/src/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression.substitute (build/cythonized/sage/symbolic/expression.cpp:32130)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   5174\u001b[0m         \u001b[0msig_on\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   5175\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 5176\u001b[0;31m             \u001b[0mres\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_gobj\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs_map\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msmap\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   5177\u001b[0m         \u001b[0;32mfinally\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   5178\u001b[0m             \u001b[0msig_off\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mValueError\u001b[0m: power::eval(): division by zero"
     ]
    }
   ],
   "source": [
    "# x is defined as a symbolic variable by default by Sage so we do not need var('x')\n",
    "f = (1+x)^(1/x)\n",
    "f.subs(x=0) # this will give you an error message"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "BUT: If you are intersted in the \"Art of dividing by zero\" talk to Professor Warwick Tucker in Maths Department!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "You can get some idea of what is going on with two plots on different scales"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
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6a+tW/fkE/flMZGenVK9eqe8zfnz6MRw+rI/36qXUr7/qOsXHR6l//tHHvb318d69dYzf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\n",
      "text/plain": [
       "Graphics Array of size 1 x 2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = (1+x)^(1/x)\n",
    "n1=5\n",
    "p1=plot(f.subs(p=1), (x, 0.001, n1), axes_labels=('x',f)) # main plot\n",
    "t1 = text(\"Large scale plot\", (n1/2,e), rgbcolor='blue',fontsize=10) \n",
    "n2=0.1\n",
    "p2=plot(f.subs(p=1), (x, 0.0000001, n2), axes_labels=('x',f)) # main plot\n",
    "p2+=line([(0,e),(n2,e)],linestyle=':') # add a dotted line at height e\n",
    "t2 = text(\"Small scale plot\", (n2/2,e+.01), rgbcolor='blue',fontsize=10) \n",
    "show(graphics_array((p1+t1,p2+t2)),figsize=[6,3]) # show the plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "all this has been laying the groundwork for the topic of real interest to us ...\n",
    "\n",
    "# Limit of a Sequence of Random Variables\n",
    "\n",
    "We want to be able to say things like $\\underset{i \\rightarrow \\infty}{\\lim} X_i = X$ in some sensible way.  $X_i$ are some random variables, $X$ is some 'limiting random variable', but what do we mean by 'limiting random variable'?\n",
    "\n",
    "To help us, lets introduce a very very simple random variable, one that puts all its mass in one place.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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mRsnJyR0RIvxATY1UUWFaAIGDHIH2UF0tVVWZFmhOJ7sDQPtzuVyyLOuk/bKz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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "Graphics Array of size 1 x 2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta = 2.0\n",
    "show(graphics_array((pmfPointMassPlot(theta),cdfPointMassPlot(theta))),\\\n",
    "     figsize=[8,2]) # show the plots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "This is known as the $Point\\,Mass(\\theta)$ random variable, $\\theta \\in \\mathbb(R)$:  the density $f(x)$ is 1 if $x=\\theta$ and 0 everywhere else\n",
    "\n",
    "$$\n",
    "f(x;\\theta) =\n",
    "\\begin{cases}\n",
    "0 & \\text{ if  }  x \\neq \\theta \\\\\n",
    "1 & \\text{ if  } x = \\theta\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "$$\n",
    "F(x;\\theta) =\n",
    "\\begin{cases}\n",
    "0 & \\text{ if  } x < \\theta \\\\\n",
    "1 & \\text{ if  } x \\geq \\theta\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "So, if we had some sequence $\\{\\theta_i\\}_{i=1}^\\infty$ and $\\underset{i \\rightarrow \\infty}{\\lim} \\theta_i = \\theta$\n",
    "\n",
    "and we had a sequence of random variables $X_i \\sim Point\\,Mass(\\theta_i)$, $i = 1, 2, 3, \\ldots$\n",
    "\n",
    "then we could talk about a limiting random variable as $X \\sim Point\\,Mass(\\theta)$:\n",
    "\n",
    "i.e., we could talk about $\\underset{i \\rightarrow \\infty}{\\lim} X_i = X$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "Graphics object consisting of 202 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# mock up a picture of a sequence of point mass rvs converging on theta = 0\n",
    "ptsize = 20\n",
    "i = 1\n",
    "theta_i = 1/i\n",
    "p = points((theta_i,1), rgbcolor=\"blue\", pointsize=ptsize)\n",
    "p += line([(theta_i,0),(theta_i,1)], rgbcolor=\"blue\", linestyle=':')\n",
    "while theta_i > 0.01:\n",
    "    i+=1\n",
    "    theta_i = 1/i\n",
    "    p += points((theta_i,1), rgbcolor=\"blue\", pointsize=ptsize)\n",
    "    p += line([(theta_i,0),(theta_i,1)], rgbcolor=\"blue\", linestyle=':')\n",
    "p += points((0,1), rgbcolor=\"red\", pointsize=ptsize)\n",
    "p += line([(0,0),(0,1)], rgbcolor=\"red\", linestyle=':')\n",
    "p.show(xmin=-1, xmax = 2, ymin=0, ymax = 1.1, axes=false, gridlines=[None,[0]], \\\n",
    "       figsize=[7,2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Now, we want to generalise this notion of a limit to other random variables (that are not necessarily $Point\\,Mass(\\theta_i)$ RVs)\n",
    "\n",
    "What about one many of you will be familiar with - the 'bell-shaped curve' \n",
    "\n",
    "## The $Gaussian(\\mu, \\sigma^2)$ or $Normal(\\mu, \\sigma^2)$ RV?\n",
    "\n",
    "The probability density function (PDF) $f(x)$ is given by\n",
    "\n",
    "$$\n",
    "f(x ;\\mu, \\sigma) = \\displaystyle\\frac{1}{\\sigma\\sqrt{2\\pi}}\\exp\\left(\\frac{-1}{2\\sigma^2}(x-\\mu)^2\\right)\n",
    "$$\n",
    "\n",
    "The two parameters, $\\mu$ and $\\sigma$, are sometimes referred to as the location and scale parameters.\n",
    "\n",
    "To see why this is, use the interactive plot below to have a look at what happens to the shape of the density function $f(x)$ when you change $\\mu$ or increase or decrease $\\sigma$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "d5adb3cbf63e4596a96e9a5d4f05de2b"
      }
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@interact\n",
    "def _(my_mu=input_box(0, label='mu') ,my_sigma=input_box(1,label='sigma')):\n",
    "    '''Interactive function to plot the normal pdf and ecdf.'''\n",
    "    \n",
    "    if my_sigma > 0:\n",
    "        html('<h4>Normal('+str(my_mu)+','+str(my_sigma)+'<sup>2</sup>)</h4>')\n",
    "        var('mu sigma')\n",
    "        f = (1/(sigma*sqrt(2.0*pi)))*exp(-1.0/(2*sigma^2)*(x - mu)^2)\n",
    "        p1=plot(f.subs(mu=my_mu,sigma=my_sigma), (x, my_mu - 3*my_sigma - 2, my_mu + 3*my_sigma + 2), axes_labels=('x','f(x)'))\n",
    "        show(p1,figsize=[8,3])\n",
    "    else:\n",
    "        print \"sigma must be greater than 0\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Consider the sequence of random variables $X_1, X_2, X_3, \\ldots$, where\n",
    "\n",
    "- $X_1 \\sim Normal(0, 1)$\n",
    "- $X_2 \\sim Normal(0, \\frac{1}{2})$\n",
    "- $X_3 \\sim Normal(0, \\frac{1}{3})$\n",
    "- $X_4 \\sim Normal(0, \\frac{1}{4})$\n",
    "- $\\vdots$\n",
    "- $X_i \\sim Normal(0, \\frac{1}{i})$\n",
    "- $\\vdots$\n",
    "\n",
    "We can use the animation below to see how the PDF $f_{i}(x)$ looks as we move through the sequence of $X_i$ (the animation only goes to $i = 25$, $\\sigma = 0.04$ but you get the picture ...)\n",
    "\n",
    "<table style=\"width:100%\">\n",
    "  <tr>\n",
    "    <th>Normal curve animation, looping through $\\sigma = \\frac{1}{i}$ for $i = 1, \\dots, 25$</th> \n",
    "  </tr>\n",
    "<tr>\n",
    "    <th><img src=\"images/normalDecreasing.gif\" width=300></th> \n",
    "  </tr>\n",
    "</table>\n",
    "\n",
    "We can see that the probability mass of $X_i \\sim Normal(0, \\frac{1}{i})$ increasingly concentrates about 0 as $i \\rightarrow \\infty$ and $\\frac{1}{i} \\rightarrow 0$\n",
    "\n",
    "Does this mean that $\\underset{i \\rightarrow \\infty}{\\lim} X_i = Point\\,Mass(0)$?\n",
    "\n",
    "No, because for any $i$, however large, $P(X_i = 0) = 0$ because $X_i$ is a continuous RV (for any continous RV $X$, for any $x \\in \\mathbb{R}$, $P(X=x) = 0$).\n",
    "\n",
    "So, we need to refine our notions of convergence when we are dealing with random variables\n",
    "\n",
    "# Convergence in Distribution\n",
    "\n",
    "Let $X_1, X_2, \\ldots$ be a sequence of random variables and let $X$ be another random variable.  Let $F_i$ denote the distribution function (DF) of $X_i$ and let $F$ denote the distribution function of $X$.\n",
    "\n",
    "Now, if for any real number $t$ at which $F$ is continuous,\n",
    "\n",
    "$$\\underset{i \\rightarrow \\infty}{\\lim} F_i(t) = F(t)$$\n",
    "\n",
    "(in the sense of the convergence or limits of functions we talked about earlier)\n",
    "\n",
    "Then we can say that the sequence or RVs $X_i$, $i = 1, 2, \\ldots$ **converges to $X$ in distribution** and write $X_i \\overset{d}{\\rightarrow} X$.\n",
    "\n",
    "An equivalent way of defining convergence in distribution is to go right back to the meaning of the probabilty space 'under the hood' of a random variable, a random variable $X$ as a mapping from the sample space $\\Omega$ to the real line ($X: \\Omega \\rightarrow \\mathbb{R}$), and the sample points or outcomes in the sample space, the $\\omega \\in \\Omega$.  For $\\omega \\in \\Omega$, $X(\\omega)$ is the mapping of $\\omega$ to the real line $\\mathbb{R}$.  We could look at the set of $\\omega$ such that $X(\\omega) \\leq t$, i.e. the set of $\\omega$ that map to some value on the real line less than or equal to $t$, $\\{\\omega: X(\\omega) \\leq t \\}$. \n",
    "\n",
    "Saying that for any $t \\in \\mathbb{R}$, $\\underset{i \\rightarrow \\infty}{\\lim} F_i(t) = F(t)$ is the equivalent of saying that for any $t \\in \\mathbb{R}$, \n",
    "\n",
    "$$\\underset{i \\rightarrow \\infty}{\\lim} P\\left(\\{\\omega:X_i(\\omega) \\leq t \\}\\right) = P\\left(\\{\\omega: X(\\omega) \\leq t\\right)$$\n",
    "\n",
    "Armed with this, we can go back to our sequence of $Normal$ random variables  $X_1, X_2, X_3, \\ldots$, where\n",
    "\n",
    "- $X_1 \\sim Normal(0, 1)$\n",
    "- $X_2 \\sim Normal(0, \\frac{1}{2})$\n",
    "- $X_3 \\sim Normal(0, \\frac{1}{3})$\n",
    "- $X_4 \\sim Normal(0, \\frac{1}{4})$\n",
    "- $\\vdots$\n",
    "- $X_i \\sim Normal(0, \\frac{1}{i})$\n",
    "- $\\vdots$\n",
    "\n",
    "and let $X \\sim Point\\,Mass(0)$,\n",
    "\n",
    "and say that the $X_i$ **converge in distribution** to the $x \\sim Point\\,Mass$ RV $X$,\n",
    "\n",
    "$$X_i \\overset{d}{\\rightarrow} X$$\n",
    "\n",
    "What we are saying with convergence in distribution, informally, is that as $i$ increases, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by the limiting random variable.  In this case, as $i$ increases, the $Point\\,Mass(0)$ is becoming a better and better model for the next outcome of a random experiment with outcomes $\\sim Normal(0,\\frac{1}{i})$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
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WM7LlE2sACAYhjYd0t0JWRKQYClkRkSKlB+9a4PaamG6L8NiqPZ4XHluFYRqEvFkSBMj0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\n/MHjOez6j7zp+MYv/YrcQJzJv/j8zp/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4HW4aKhqwldmoO1FHdV41Nfk1VOdVe767mtRo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\nQRCuSP8fKd8nH78OXJsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "Graphics object consisting of 14 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# mock up a picture of a sequence of converging normal distributions\n",
    "my_mu = 0\n",
    "upper = my_mu + 5; lower = -upper;     # limits for plot\n",
    "var('mu sigma')\n",
    "stop_i = 12\n",
    "html('<h4>N(0,1) to N(0, 1/'+str(stop_i)+')</h4>')\n",
    "f = (1/(sigma*sqrt(2.0*pi)))*exp(-1.0/(2*sigma^2)*(x - mu)^2)\n",
    "p=plot(f.subs(mu=my_mu,sigma=1.0), (x, lower, upper), rgbcolor = (0,0,1))\n",
    "for i in range(2, stop_i, 1): # just do a few of them\n",
    "    shade = 1-11/i # make them different colours\n",
    "    p+=plot(f.subs(mu=my_mu,sigma=1/i), (x, lower, upper), rgbcolor = (1-shade, 0, shade))\n",
    "textOffset = -0.2 # offset for placement of text -  may need adjusting \n",
    "p+=text(\"0\",(0,textOffset),fontsize = 10, rgbcolor='grey') \n",
    "p+=text(str(upper.n(digits=2)),(upper,textOffset),fontsize = 10, rgbcolor='grey') \n",
    "p+=text(str(lower.n(digits=2)),(lower,textOffset),fontsize = 10, rgbcolor='grey') \n",
    "p.show(axes=false, gridlines=[None,[0]], figsize=[7,3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "#### There is an interesting point to note about this convergence: \n",
    "\n",
    "We have said that the $X_i \\sim Normal(0,\\frac{1}{i})$ with distribution functions $F_i$ converge in distribution to $X \\sim Point\\,Mass(0)$ with distribution function $F$, which means that we must be able to show that for any real number $t$ at which $F$ is continuous,\n",
    "\n",
    "$$\\underset{i \\rightarrow \\infty}{\\lim} F_i(t) = F(t)$$\n",
    "\n",
    "Note that for any of the $X_i \\sim Normal(0, \\frac{1}{i})$, $F_i(0) = \\frac{1}{2}$, and also note that for $X \\sim Point,Mass(0)$, $F(0) = 1$, so clearly $F_i(0) \\neq F(0)$.  \n",
    "\n",
    "What has gone wrong?  \n",
    "\n",
    "Nothing:  we said that we had to be able to show that $\\underset{i \\rightarrow \\infty}{\\lim} F_i(t) = F(t)$ for any $t \\in \\mathbb{R}$ at which $F$ is continuous, but the $Point\\,Mass(0)$ distribution function $F$ is not continous at 0!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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vX6/u3btLkvbv36927eo+1n/4wx9UXV2tjIwMWzvz5s3T3LlzG9zGzp11cdSo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\n",
      "text/plain": [
       "Graphics Array of size 1 x 2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta = 0.0\n",
    "# show the plots\n",
    "show(graphics_array((pmfPointMassPlot(theta),cdfPointMassPlot(theta))),figsize=[8,2]) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# Convergence in Probability\n",
    "\n",
    "Let $X_1, X_2, \\ldots$ be a sequence of random variables and let $X$ be another random variable.  Let $F_i$ denote the distribution function (DF) of$X_i$ and let $F$ denote the distribution function of $X$.\n",
    "\n",
    "Now, if for any real number $\\varepsilon > 0$,\n",
    "\n",
    "$$\\underset{i \\rightarrow \\infty}{\\lim} P\\left(|X_i - X| > \\varepsilon\\right) = 0$$\n",
    "\n",
    "Then we can say that the sequence $X_i$, $i = 1, 2, \\ldots$ **converges to $X$ in probability** and write $X_i \\overset{P}{\\rightarrow} X$.\n",
    "\n",
    "Or, going back again to the probability space 'under the hood' of a random variable,  we could look the way the $X_i$ maps each outcome $\\omega \\in \\Omega$, $X_i(\\omega)$, which is some point on the real line, and compare this to mapping $X(\\omega)$.   \n",
    "\n",
    "Saying that for any $\\varepsilon \\in \\mathbb{R}$, $\\underset{i \\rightarrow \\infty}{\\lim} P\\left(|X_i - X| > \\varepsilon\\right) = 0$ is the equivalent of saying that for any $\\varepsilon \\in \\mathbb{R}$, \n",
    "\n",
    "$$\\underset{i \\rightarrow \\infty}{\\lim} P\\left(\\{\\omega:|X_i(\\omega) - X(\\omega)| > \\varepsilon \\}\\right) = 0$$\n",
    "\n",
    "Informally, we are saying $X$ is a limit in probabilty if, by going far enough into the sequence $X_i$, we can ensure that the mappings $X_i(\\omega)$ and $X(\\omega)$ will be arbitrarily close to each other on the real line for all $\\omega \\in \\Omega$.\n",
    "\n",
    "**Note** that convergence in distribution is implied by convergence in probability: convergence in distribution is the weakest form of convergence; any sequence of RV's that converges in probability to some RV $X$ also converges in distribution to $X$ (but not necessarily vice versa).  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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+gJA/dNDnDpeDgQ+OZesXWeQu3kHr83pSsHQ7iZlt8VUGCRhuQpiY1ZUneqsi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\nRLYMosuEJ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      "text/plain": [
       "Graphics object consisting of 25 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# mock up a picture of a sequence of converging normal distributions\n",
    "my_mu = 0\n",
    "var('mu sigma')\n",
    "upper = 0.2; lower = -upper\n",
    "i = 20 # start part way into the sequence\n",
    "lim = 100 # how far to go\n",
    "stop_i = 12\n",
    "html('<h4>N(0,1/'+str(i)+') to N(0, 1/'+str(lim)+')</h4>')\n",
    "f = (1/(sigma*sqrt(2.0*pi)))*exp(-1.0/(2*sigma^2)*(x - mu)^2)\n",
    "p=plot(f.subs(mu=my_mu,sigma=1.0/i), (x, lower, upper), rgbcolor = (0,0,1))\n",
    "for j in range(i, lim+1, 4): # just do a few of them\n",
    "    shade = 1-(j-i)/(lim-i) # make them different colours\n",
    "    p+=plot(f.subs(mu=my_mu,sigma=1/j), (x, lower,upper), rgbcolor = (1-shade, 0, shade))\n",
    "textOffset = -1.5 # offset for placement of text -  may need adjusting \n",
    "p+=text(\"0\",(0,textOffset),fontsize = 10, rgbcolor='grey') \n",
    "p+=text(str(upper.n(digits=2)),(upper,textOffset),fontsize = 10, rgbcolor='grey') \n",
    "p+=text(str(lower.n(digits=2)),(lower,textOffset),fontsize = 10, rgbcolor='grey') \n",
    "p.show(axes=false, gridlines=[None,[0]], figsize=[7,3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "For our sequence of $Normal$ random variables  $X_1, X_2, X_3, \\ldots$, where\n",
    "\n",
    "- $X_1 \\sim Normal(0, 1)$\n",
    "- $X_2 \\sim Normal(0, \\frac{1}{2})$\n",
    "- $X_3 \\sim Normal(0, \\frac{1}{3})$\n",
    "- $X_4 \\sim Normal(0, \\frac{1}{4})$\n",
    "- $\\vdots$\n",
    "- $X_i \\sim Normal(0, \\frac{1}{i})$\n",
    "- $\\vdots$\n",
    "\n",
    "and $X \\sim Point\\,Mass(0)$,\n",
    "\n",
    "It can be shown that the $X_i$ converge in probability to $X \\sim Point\\,Mass(0)$ RV $X$,\n",
    "\n",
    "$$X_i \\overset{P}{\\rightarrow} X$$\n",
    "\n",
    "(the formal proof of this involves Markov's Inequality, which is beyond the scope of this course). \n",
    "\n",
    "# Some Basic Limit Laws in Statistics\n",
    "\n",
    "Intuition behind Law of Large Numbers and Central Limit Theorem\n",
    "\n",
    "Take a look at the Khan academy videos on the Law of Large Numbers and the Central Limit Theorem. This will give you a working idea of these theorems.  In the sequel, we will strive for a deeper understanding of these theorems on the basis of the two notions of convergence of sequences of random variables we just saw.\n",
    "   \n",
    "\n",
    "## Weak Law of Large Numbers\n",
    "\n",
    "Remember that a statistic is a random variable, so a sample mean is a random variable. If we are given a sequence of independent and identically distributed RVs, $X_1,X_2,\\ldots \\overset{IID}{\\sim} X_1$, then we can also think of a sequence of random variables $\\overline{X}_1, \\overline{X}_2, \\ldots, \\overline{X}_n, \\ldots$ ($n$ being the sample size). \n",
    "\n",
    "Since $X_1, X_2, \\ldots$ are $IID$, they all have the same expection, say $E(X_1)$ by convention.\n",
    "\n",
    "If $E(X_1)$ exists, then the sample mean $\\overline{X}_n$ converges in probability to $E(X_1)$ (i.e., to the expectatation of any one of the individual RVs):\n",
    "\n",
    "$$\n",
    "\\text{If} \\quad X_1,X_2,\\ldots \\overset{IID}{\\sim} X_1 \\ \\text{and if } \\ E(X_1) \\ \\text{exists, then } \\ \\overline{X}_n \\overset{P}{\\rightarrow} E(X_1) \\ .\n",
    "$$\n",
    "\n",
    "Going back to our definition of convergence in probability, we see that this means that for any real number $\\varepsilon > 0$, $\\underset{n \\rightarrow \\infty}{\\lim} P\\left(|\\overline{X}_n - E(X_1)| > \\varepsilon\\right) = 0$\n",
    "\n",
    "Informally, this means that means that, by taking larger and larger samples we can make the probability that the average of the observations is more than $\\varepsilon$ away from the expected value get smaller and smaller.\n",
    "\n",
    "Proof of this is beyond the scope of this course, but we have already seen it in action when we looked at the $Bernoulli$ running means.  Have another look, this time with only one sequence of running means.  You can increase $n$, the sample size, and change $\\theta$.  Note that the seed for the random number generator is also under your control.   This means that you can get replicable samples:  in particular, in this interact, when you increase the sample size it looks as though you are just adding more to an existing sample rather than starting from scratch with a new one.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "d636c4d2cbb4400190a8e5b7a1a41d38"
      }
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@interact\n",
    "def _(nToGen=slider(1,1500,1,100,label='n'),my_theta=input_box(0.3,label='theta'),rSeed=input_box(1234,label='random seed')):\n",
    "    '''Interactive function to plot running mean for a Bernoulli with specified n, theta and random number seed.'''\n",
    "    \n",
    "    if my_theta >= 0 and my_theta <= 1:\n",
    "        html('<h4>Bernoulli('+str(my_theta.n(digits=2))+')</h4>')\n",
    "        xvalues = range(1, nToGen+1,1)\n",
    "        bRunningMeans = bernoulliRunningMeans(nToGen, myTheta=my_theta, mySeed=rSeed)\n",
    "        pts = zip(xvalues, bRunningMeans)\n",
    "        p = line(pts, rgbcolor = (0,0,1))\n",
    "        p+=line([(0,my_theta),(nToGen,my_theta)],linestyle=':',rgbcolor='grey')\n",
    "        show(p, figsize=[5,3], axes_labels=['n','sample mean'],ymax=1)\n",
    "    else:\n",
    "        print 'Theta must be between 0 and 1'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# Central Limit Theorem\n",
    "\n",
    "You have probably all heard of the Central Limit Theorem before, but now we can relate it to our definition of convergence in distribution.  \n",
    "\n",
    "Let $X_1,X_2,\\ldots \\overset{IID}{\\sim} X_1$ and suppose $E(X_1)$ and $V(X_1)$ both exist,\n",
    "\n",
    "then\n",
    "\n",
    "$$\n",
    "\\overline{X}_n = \\frac{1}{n} \\sum_{i=1}^n X_i  \\overset{d}{\\rightarrow} X \\sim Normal \\left(E(X_1),\\frac{V(X_1)}{n} \\right)\n",
    "$$\n",
    "\n",
    "And remember $Z \\sim Normal(0,1)$?\n",
    "\n",
    "Consider $Z_n := \\displaystyle\\frac{\\overline{X}_n-E(\\overline{X}_n)}{\\sqrt{V(\\overline{X}_n)}} = \\displaystyle\\frac{\\sqrt{n} \\left( \\overline{X}_n -E(X_1) \\right)}{\\sqrt{V(X_1)}}$\n",
    "\n",
    "If $\\overline{X}_n = \\displaystyle\\frac{1}{n} \\displaystyle\\sum_{i=1}^n X_i  \\overset{d}{\\rightarrow}  X \\sim Normal \\left(E(X_1),\\frac{V(X_1)}{n} \\right)$, then $\\overline{X}_n -E(X_1)  \\overset{d}{\\rightarrow}  X-E(X_1) \\sim Normal \\left( 0,\\frac{V(X_1)}{n} \\right)$\n",
    "\n",
    "and $\\sqrt{n} \\left( \\overline{X}_n -E(X_1) \\right)  \\overset{d}{\\rightarrow}  \\sqrt{n} \\left( X-E(X_1) \\right) \\sim Normal \\left( 0,V(X_1) \\right)$\n",
    "\n",
    "so $Z_n := \\displaystyle \\frac{\\overline{X}_n-E(\\overline{X}_n)}{\\sqrt{V(\\overline{X}_n)}} = \\displaystyle\\frac{\\sqrt{n} \\left( \\overline{X}_n -E(X_1) \\right)}{\\sqrt{V(X_1)}}  \\overset{d}{\\rightarrow}  Z \\sim Normal \\left( 0,1 \\right)$\n",
    "\n",
    "Thus, for sufficiently large $n$ (say $n>30$), probability statements about $\\overline{X}_n$ can be approximated using the $Normal$ distribution. \n",
    "\n",
    "The beauty of the CLT, as you have probably seen from other courses, is that $\\overline{X}_n \\overset{d}{\\rightarrow} Normal \\left( E(X_1), \\frac{V(X_1)}{n} \\right)$ does not require the $X_i$ to be normally distributed. \n",
    "\n",
    "We can try this with our $Bernoulli$ RV generator.  First, a small number of samples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[7/10, 9/10, 3/10, 4/5, 2/5]"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta, n, samples = 0.6, 10, 5 # concise way to set some variable values\n",
    "sampleMeans=[] # empty list\n",
    "for i in range(0, samples, 1):  # loop \n",
    "    thisMean = QQ(sum(bernoulliSample(n, theta)))/n # get a sample and find the mean\n",
    "    sampleMeans.append(thisMean) # add mean to the list of means\n",
    "sampleMeans    # disclose the sample means"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "You can use the interactive plot to increase the number of samples and make a histogram of the sample means.  According to the CLT, for lots of reasonably-sized samples we should get a nice symmetric bell-curve-ish histogram centred on $\\theta$.  You can adjust the number of bins in the histogram as well as the number of samples, sample size, and $\\theta$.   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
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       "model_id": "3092264820fb4d06adacf64b86dc9b45"
      }
     },
     "metadata": {},
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    }
   ],
   "source": [
    "import pylab\n",
    "@interact\n",
    "def _(samples=slider(1,3000,1,100,label='number of samples'), nToGen=slider(1,1500,1,100,label='sample size n'),my_theta=input_box(0.3,label='theta'),Bins=5):\n",
    "    '''Interactive function to plot distribution of sample means for a Bernoulli process.'''\n",
    "    \n",
    "    if my_theta >= 0 and my_theta <= 1 and samples > 0:\n",
    "        sampleMeans=[] # empty list\n",
    "        for i in range(0, samples, 1):          \n",
    "            thisMean = RR(sum(bernoulliSample(nToGen, my_theta)))/nToGen\n",
    "            sampleMeans.append(thisMean)\n",
    "        pylab.clf() # clear current figure\n",
    "        n, bins, patches = pylab.hist(sampleMeans, Bins, normed=true) \n",
    "        pylab.ylabel('normalised count')\n",
    "        pylab.title('Normalised histogram for Bernoulli sample means')\n",
    "        pylab.savefig('myHist') # to actually display the figure\n",
    "        pylab.show()\n",
    "        #show(p, figsize=[5,3], axes_labels=['n','sample mean'],ymax=1)\n",
    "    else:\n",
    "        print 'Theta must be between 0 and 1, and samples > 0'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Increase the sample size and the numbe rof bins in the above interact and see if the histograms of the sample means are looking more and more normal as the CLT would have us believe."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "But although the $X_i$ do not have to be $\\sim Normal$ for $\\overline{X}_n = \\overset{d}{\\rightarrow} X \\sim Normal\\left(E(X_1),\\frac{V(X_1)}{n} \\right)$, remember that we said \"Let $X_1,X_2,\\ldots \\overset{IID}{\\sim} X_1$ and suppose $E(X_1)$ and $V(X_1)$ both exist\", then,\n",
    "\n",
    "$$\n",
    "\\overline{X}_n = \\frac{1}{n} \\sum_{i=1}^n X_i  \\overset{d}{\\rightarrow} X \\sim Normal \\left(E(X_1),\\frac{V(X_1)}{n} \\right)\n",
    "$$\n",
    "\n",
    "This is where is all goes horribly wrong for the standard $Cauchy$ distribution (any $Cauchy$ distribution in fact):  neither the expectation nor the variance exist for this distribution.  The Central Limit Theorem cannot be applied here.  In fact, if $X_1,X_2,\\ldots \\overset{IID}{\\sim}$ standard $Cauchy$, then $\\overline{X}_n = \\displaystyle \\frac{1}{n} \\sum_{i=1}^n X_i  \\sim$ standard $Cauchy$.\n",
    "\n",
    "### YouTry\n",
    "\n",
    "Try looking at samples from two other RVs where the expectation and variance do exist, the $Uniform$ and the $Exponential$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "9a0c2d075ad34f56b689bf4894aaadec"
      }
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import pylab\n",
    "@interact\n",
    "def _(samples=input_box(100,label='number of samples'), nToGen=slider(1,1500,1,100,label='sample size n'),my_theta1=input_box(2,label='theta1'),my_theta2=input_box(4,label='theta1'),Bins=5):\n",
    "    '''Interactive function to plot distribution of sample means for a Uniform(theta1, theta2) process.'''\n",
    "    \n",
    "    if (my_theta1 < my_theta2) and samples > 0:\n",
    "        sampleMeans=[] # empty list\n",
    "        for i in range(0, samples, 1):\n",
    "            \n",
    "            thisMean = RR(sum(uniformSample(nToGen, my_theta1, my_theta2)))/nToGen\n",
    "            sampleMeans.append(thisMean)\n",
    "        pylab.clf() # clear current figure\n",
    "        n, bins, patches = pylab.hist(sampleMeans, Bins, normed=true) \n",
    "        pylab.ylabel('normalised count')\n",
    "        pylab.title('Normalised histogram for Uniform sample means')\n",
    "        pylab.savefig('myHist') # to actually display the figure\n",
    "        pylab.show()\n",
    "        #show(p, figsize=[5,3], axes_labels=['n','sample mean'],ymax=1)\n",
    "    else:\n",
    "        print 'theta1 must be less than theta2, and samples > 0'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "40ab507b87824dd38b17b491f6216b00"
      }
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import pylab\n",
    "@interact\n",
    "def _(samples=input_box(100,label='number of samples'), nToGen=slider(1,1500,1,100,label='sample size n'),my_lambda=input_box(2,label='lambda'),Bins=5):\n",
    "    '''Interactive function to plot distribution of sample means for an Exponential(lambda) process.'''\n",
    "    \n",
    "    if my_lambda > 0 and samples > 0:\n",
    "        sampleMeans=[] # empty list\n",
    "        for i in range(0, samples, 1):\n",
    "            \n",
    "            thisMean = RR(sum(exponentialSample(nToGen, my_lambda)))/nToGen\n",
    "            sampleMeans.append(thisMean)\n",
    "        pylab.clf() # clear current figure\n",
    "        n, bins, patches = pylab.hist(sampleMeans, Bins, normed=true) \n",
    "        pylab.ylabel('normalised count')\n",
    "        pylab.title('Normalised histogram for Exponential sample means')\n",
    "        pylab.savefig('myHist') # to actually display the figure\n",
    "        pylab.show()\n",
    "        #show(p, figsize=[5,3], axes_labels=['n','sample mean'],ymax=1)\n",
    "    else:\n",
    "        print 'lambda must be greater than 0, and samples > 0'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### YouTry Later\n",
    "\n",
    "Python's `random` for sampling and sequence manipulation\n",
    "\n",
    "The Python `random` module, available in SageMath, provides a useful way of taking samples if you have already generated a 'population' to sample from, or otherwise playing around with the elements in a sequence.  See http://docs.python.org/library/random.html for more details.  Here we will try a few of them.\n",
    "\n",
    "The aptly-named sample function allows us to take a sample of a specified size from a sequence.  We will use a list as our sequence:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[37, 51, 2, 35, 19, 59, 40, 96, 14, 80]"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pop = range(1, 101, 1) # make a population\n",
    "sample(pop, 10) # sample 10 elements from it at random"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Each call to sample will select unique elements in the list (note that 'unique' here means that it will not select the element at any particular position in the list more than once, but if there are duplicate elements in the list, such as with a list [1,2,4,2,5,3,1,3], then you may well get any of the repeated elements in your sample more than once).  sample samples with replacement, which means that repeated calls to sample may give you samples with the same elements in."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\n"
     ]
    }
   ],
   "source": [
    "popWithDuplicates = range(1, 11, 1)*4 # make a population with repeated elements\n",
    "print(popWithDuplicates)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[8, 1, 8, 4, 2, 4, 9, 2, 5, 2]\n",
      "[7, 1, 6, 4, 1, 5, 2, 6, 3, 5]\n",
      "[10, 1, 8, 4, 1, 6, 3, 3, 2, 1]\n",
      "[7, 9, 9, 3, 10, 7, 2, 1, 6, 5]\n",
      "[8, 1, 7, 1, 5, 2, 4, 6, 4, 9]\n"
     ]
    }
   ],
   "source": [
    "for i in range (5):\n",
    "    print sample(popWithDuplicates, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Try experimenting with choice, which allows you to select one element at random from a sequence, and shuffle, which shuffles the sequence in place (i.e, the ordering of the sequence itself is changed rather than you being given a re-ordered copy of the list).  It is probably easiest to use lists for your sequences.  See how `shuffle` is creating permutations of the list.  You could use `sample` and `shuffle` to emulate *permuations of k objects out of n* ...\n",
    "\n",
    "You may need to check the documentation to see how use these functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "?sample"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "?shuffle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "?choice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": []
  }
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