From a8d49f0e9faee9f6ad83278113200aa4b54a1146 Mon Sep 17 00:00:00 2001 From: Peter Norvig Date: Wed, 18 Mar 2026 11:26:16 -0700 Subject: [PATCH] Add files via upload --- ipynb/Mean Misanthrope Density.ipynb | 804 ++++++++++++--------------- ipynb/houses4.jpg | Bin 0 -> 122667 bytes 2 files changed, 367 insertions(+), 437 deletions(-) create mode 100644 ipynb/houses4.jpg diff --git a/ipynb/Mean Misanthrope Density.ipynb b/ipynb/Mean Misanthrope Density.ipynb index 73a1b44..07a3bf8 100644 --- a/ipynb/Mean Misanthrope Density.ipynb +++ b/ipynb/Mean Misanthrope Density.ipynb @@ -4,24 +4,29 @@ "cell_type": "markdown", "metadata": {}, "source": [ + "
Peter Norvig
2016
\n", + "\n", "# The Puzzle of the Misanthropic Neighbors" ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ + "In this notebook we look at two puzzles, and show how they can be solved by a combination of computational and mathematical thinking.\n", + "\n", "Consider [this puzzle](http://fivethirtyeight.com/features/can-you-solve-the-puzzle-of-your-misanthropic-neighbors/) from [The Riddler](http://fivethirtyeight.com/tag/the-riddler/):\n", " \n", "> *The misanthropes are coming. Suppose there is a row of some number, N, of houses in a new, initially empty development. Misanthropes are moving into the development one at a time and selecting a house at random from those that have nobody in them and nobody living next door. They keep on coming until no acceptable houses remain. At most, one out of two houses will be occupied; at least one out of three houses will be. But what’s the expected fraction of occupied houses as the development gets larger, that is, as N goes to infinity?*\n", "\n", - "# Consider *N*=4 Houses\n", "\n", - "![N houses](http://www.publicdomainpictures.net/pictures/180000/nahled/street-doodle.jpg)\n", + "\n", + "\n", "\n", "To make sure we understand the problem, let's try a simple example, with *N*=4 houses. We will represent the originally empty row of four houses by four dots:\n", "\n", - " ....\n", + " Houses: ....\n", " \n", "Now the first person chooses one of the four houses (which are all acceptable). We'll indicate an occupied house by a `1`, so the four equiprobable choices are:\n", "\n", @@ -37,11 +42,15 @@ " Choice 2: .010\n", " Choice 3: ..01\n", " \n", - "In all four cases, a second occupant has a place to move in, but then there is no place for a third occupant. \n", - "The occupancy is 2 in all cases, and thus the *expected occpancy* is 2. The occupancy fraction, or *density*, is 2/*N* = 2/4 = 1/2.\n", - " \n", + "In all four cases, a second occupant has a place to move in, but then there is no place for a third occupant:\n", "\n", - "# Consider *N*=7 Houses\n", + " Choice 0: 10.. → 1010 or 1001\n", + " Choice 1: 010. → 0101\n", + " Choice 2: .010 → 1010\n", + " Choice 3: ..01 → 0101 or 1001\n", + "\n", + "The occupancy is 2 in all cases, and thus the *expected occupancy* is 2. The occupancy fraction, or *density*, is 2/*N* = 2/4 = 1/2.\n", + " \n", "\n", "With *N*=7 houses, there are 7 equiprobable choices for the first occupant:\n", "\n", @@ -63,19 +72,19 @@ " Choice 5: ....010 runs = (4, 0) \n", " Choice 6: .....01 runs = (5, 0) \n", " \n", - "# Defining `occ(n)`\n", + "# Defining occupancy(n)\n", "\n", - "This gives me a key hint as to how to analyze the problem. I'll define `occ(n)` to be the expected number of occupied houses in a row of `n` houses. So:\n", + "This gives me a key hint as to how to analyze the problem. I'll define `occupancy(n)` to be the expected number of occupied houses in a row of `n` houses. So:\n", "\n", - " Choice 0: 10..... runs = (0, 5); occupancy = occ(0) + 1 + occ(5)\n", - " Choice 1: 010.... runs = (0, 4); occupancy = occ(0) + 1 + occ(4)\n", - " Choice 2: .010... runs = (1, 3); occupancy = occ(1) + 1 + occ(3)\n", - " Choice 3: ..010.. runs = (2, 2); occupancy = occ(2) + 1 + occ(2)\n", - " Choice 4: ...010. runs = (3, 1); occupancy = occ(3) + 1 + occ(1)\n", - " Choice 5: ....010 runs = (4, 0); occupancy = occ(4) + 1 + occ(0)\n", - " Choice 6: .....01 runs = (5, 0); occupancy = occ(5) + 1 + occ(0) \n", + " Choice 0: 10..... runs = (0, 5); occupancy = occupancy(0) + 1 + occupancy(5)\n", + " Choice 1: 010.... runs = (0, 4); occupancy = occupancy(0) + 1 + occupancy(4)\n", + " Choice 2: .010... runs = (1, 3); occupancy = occupancy(1) + 1 + occupancy(3)\n", + " Choice 3: ..010.. runs = (2, 2); occupancy = occupancy(2) + 1 + occupancy(2)\n", + " Choice 4: ...010. runs = (3, 1); occupancy = occupancy(3) + 1 + occupancy(1)\n", + " Choice 5: ....010 runs = (4, 0); occupancy = occupancy(4) + 1 + occupancy(0)\n", + " Choice 6: .....01 runs = (5, 0); occupancy = occupancy(5) + 1 + occupancy(0) \n", " \n", - "So we can say that `occ(n)` is:\n", + "So we can say that `occupancy(n)` is:\n", "\n", "- 0 when `n` is 0 (because no houses means no occupants),\n", "- 1 when `n` is 1 (because one isolated acceptable house has one occupant),\n", @@ -92,28 +101,28 @@ "outputs": [], "source": [ "from statistics import mean\n", + "from functools import cache\n", "\n", - "def occ(n):\n", - " \"The expected occupancy for a row of n houses (under misanthrope rules).\"\n", + "@cache\n", + "def occupancy(n: int) -> float:\n", + " \"\"\"The expected occupancy for a row of n houses (under misanthrope rules).\"\"\"\n", " return (0 if n == 0 else\n", " 1 if n == 1 else\n", - " mean(occ(L) + 1 + occ(R)\n", - " for (L, R) in runs(n)))\n", + " mean(occupancy(L) + 1 + occupancy(R)\n", + " for (L, R) in possible_runs(n)))\n", "\n", - "def runs(n):\n", + "def possible_runs(n: int) -> list[tuple[int, int]]:\n", " \"\"\"A list [(L, R), ...] where the i-th tuple contains the lengths of the runs\n", " of acceptable houses to the left and right of house i.\"\"\"\n", " return [(max(0, i - 1), max(0, n - i - 2))\n", - " for i in range(n)]\n", - "\n", - "def density(n): return occ(n) / n" + " for i in range(n)]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Let's check that `occ(4)` is 2, as we computed it should be, and for other small values, up to 7:" + "Let's check that `occupancy(4)` is 2, as we computed it should be, and for other small values, up to 10:" ] }, { @@ -131,7 +140,9 @@ " 4: 2,\n", " 5: 2.466666666666667,\n", " 6: 2.888888888888889,\n", - " 7: 3.323809523809524}" + " 7: 3.323809523809524,\n", + " 8: 3.7555555555555555,\n", + " 9: 4.188007054673721}" ] }, "execution_count": 2, @@ -140,43 +151,30 @@ } ], "source": [ - "{n: occ(n) for n in range(8)}" + "{n: occupancy(n) for n in range(10)}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "That looks good, although I can't prove the value above 4 are ccorrect. Now check that `runs(7)` is what we described above:" + "That looks good, although I can't prove the value above 4 are correct. Now check that `possible_runs(7)` is what we described above:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "[(0, 5), (0, 4), (1, 3), (2, 2), (3, 1), (4, 0), (5, 0)]" - ] - }, - "execution_count": 3, - "metadata": {}, - "output_type": "execute_result" - } - ], + "outputs": [], "source": [ - "runs(7)" + "assert possible_runs(7) == [(0, 5), (0, 4), (1, 3), (2, 2), (3, 1), (4, 0), (5, 0)]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# Dynamic Programming Version of `occ`\n", - "\n", - "The computation of `occ(n)` makes multiple calls to `occ(n-1), occ(n-2),` etc. To avoid re-computing the same calls over and over, we will modify `occ` to save previous results using [dynamic programming](https://en.wikipedia.org/wiki/Dynamic_programming). I could implement that in one line with the decorator [`@functools.lru_cache`](https://docs.python.org/3/library/functools.html#functools.lru_cache), but then I would have to worry about the recursion limit. Instead I will explicitly manage a list, `cache`, such that `cache[n]` holds `occ(n)`:" + "The problem asks not for the occupancy, but for the \"expected fraction of occupied houses,\" which I will call `density`:" ] }, { @@ -185,20 +183,9 @@ "metadata": {}, "outputs": [], "source": [ - "def occ(n, cache=[0, 1]):\n", - " \"The expected occupancy for a row of n houses (under misanthrope rules).\"\n", - " # Store occ(i) in cache[i] for all as-yet-uncomputed values of i up to n:\n", - " for i in range(len(cache), n+1):\n", - " cache.append(mean(cache[L] + 1 + cache[R]\n", - " for (L, R) in runs(i)))\n", - " return cache[n]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let's make sure this new version gets the same results as the old version:" + "def density(n: int) -> float: \n", + " \"\"\"Expected fraction of occupied houses.\"\"\"\n", + " return occupancy(n) / n" ] }, { @@ -209,7 +196,15 @@ { "data": { "text/plain": [ - "True" + "{1: 1.0,\n", + " 2: 0.5,\n", + " 3: 0.5555555555555556,\n", + " 4: 0.5,\n", + " 5: 0.49333333333333335,\n", + " 6: 0.48148148148148145,\n", + " 7: 0.47482993197278917,\n", + " 8: 0.46944444444444444,\n", + " 9: 0.46533411718596907}" ] }, "execution_count": 5, @@ -218,34 +213,31 @@ } ], "source": [ - "occ(4) == 2" - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "3.323809523809524" - ] - }, - "execution_count": 6, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "occ(7)" + "{n: density(n) for n in range(1, 10)}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Let's make sure the caching makes computation pretty fast the first time, and *very* fast the second time:" + "# Plotting density(n)\n", + "\n", + "The `density(n)` is around 1/2 for all values of *n* > 1. To get a better feel, let's make a plot:" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "\n", + "def plot_density(nums: range) -> float:\n", + " \"\"\"Plot density(n) for each n in `nums`, return desity(max(nums)).\"\"\"\n", + " plt.xlabel('n houses'); plt.ylabel('density(n)')\n", + " plt.plot(nums, [density(n) for n in nums], 's-')\n", + " return density(max(nums))" ] }, { @@ -253,27 +245,36 @@ "execution_count": 7, "metadata": {}, "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "CPU times: user 2.68 s, sys: 13.5 ms, total: 2.7 s\n", - "Wall time: 2.71 s\n" - ] - }, { "data": { "text/plain": [ - "864.9617138385324" + "0.4422322608865297" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" + }, + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" } ], "source": [ - "%time occ(2000)" + "plot_density(range(1, 31))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "And let's look at a wider range:" ] }, { @@ -281,75 +282,21 @@ "execution_count": 8, "metadata": {}, "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "CPU times: user 4 µs, sys: 0 ns, total: 4 µs\n", - "Wall time: 5.01 µs\n" - ] - }, { "data": { "text/plain": [ - "864.9617138385324" + "0.4328273535069354" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" - } - ], - "source": [ - "%time occ(2000)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Plotting `density(n)`\n", - "\n", - "To get a feel for the limit of `density(n)`, start by drawing a plot over some small values of `n`:" - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "metadata": {}, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "import matplotlib.pyplot as plt\n", - "plt.style.use('fivethirtyeight')\n", - "\n", - "def plot_density(ns):\n", - " \"Plot density(n) for each n in the list of numbers ns.\"\n", - " plt.xlabel('n houses'); plt.ylabel('density(n)')\n", - " plt.plot(ns, [density(n) for n in ns], 's-')\n", - " return density(ns[-1])" - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "0.4353323288377046" - ] - }, - "execution_count": 10, - "metadata": {}, - "output_type": "execute_result" }, { "data": { - "image/png": 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OwbKXLl2Co6Ojxo+TkxNOnz5t8vX0vYfYxUEEx3Yi9Oqi2UPkc0QiImGw+uTe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\nwqnLzMxESEiIdSplQxYvXqw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", "text/plain": [ - "" + "
" ] }, "metadata": {}, @@ -357,88 +304,7 @@ } ], "source": [ - "plot_density(range(1, 100))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "There is something funny going on with the first few values of `n`. Let's separately look at the first few:" - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "0.46203174603174607" - ] - }, - "execution_count": 11, - "metadata": {}, - "output_type": "execute_result" - }, - { - "data": { - "image/png": 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RE7PpJ/jatWubqw6Hp1YpTQLwhEaLv7TzkLAiIiK6GZwJpokspkRjT1AiIqfWYAB+8803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\nzZs3W7yHSqXCsmXLjNtlZWV44YUXEBM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HHntMeP3EiRPCQyXozo22ZhgTEfUHBoXx4sWL8e6778LNzQ1PPPEEUlNTYWpq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\nTIZhw4YhNjZWCHgAqKqqwl/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- "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "plot_density(range(100, 4000, 50))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The density is going down, and the curve is almost but not quite flat. " + "plot_density(range(20, 601, 20))" ] }, { @@ -450,7 +316,7 @@ "# lim n → ∞ density(n)\n", "\n", "[The puzzle](http://fivethirtyeight.com/features/can-you-solve-the-puzzle-of-your-misanthropic-neighbors/)\n", - "is to figure out the limit of `density(n)` as `n` goes to infinity. The plot above makes it look like 0.432+something, but we can't answer the question just by plotting; we'll need to switch modes from *computational* thinking to *mathematical* thinking.\n", + "is to figure out the limit of `density(n)` as `n` goes to infinity. It looks like something around 0.432, but we can't answer the question just by plotting; we'll need to switch modes from *computational* thinking to *mathematical* thinking.\n", "\n", "At this point I started playing around with `density(n)`, looking at various values, differences of values, ratios of values, and ratios of differences of values, hoping to achieve some mathematical insight. Mostly I got dead ends.\n", "But eventually I hit on something promising. I looked at the difference between density values (using the function `diff`), and particularly the difference as you double `n`:" @@ -458,7 +324,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 9, "metadata": {}, "outputs": [ { @@ -467,13 +333,15 @@ "0.0014849853757253895" ] }, - "execution_count": 13, + "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "def diff(n, m): return density(n) - density(m)\n", + "def diff(n, m) -> float: \n", + " \"\"\"The difference in density between n and m houses.\"\"\"\n", + " return density(n) - density(m)\n", "\n", "diff(100, 200)" ] @@ -487,7 +355,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 10, "metadata": {}, "outputs": [ { @@ -496,7 +364,7 @@ "0.0007424926878626947" ] }, - "execution_count": 14, + "execution_count": 10, "metadata": {}, "output_type": "execute_result" } @@ -514,7 +382,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 11, "metadata": {}, "outputs": [ { @@ -523,7 +391,7 @@ "2.0" ] }, - "execution_count": 15, + "execution_count": 11, "metadata": {}, "output_type": "execute_result" } @@ -536,47 +404,32 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Wow—not only is it *close* to twice as much, it is *exactly* twice as much (to the precision of floating point numbers). Let's try other starting values for `n`:" + "Wow—not only is it *close* to twice as much, it is *exactly* twice as much (to the precision of floating point numbers). Let's try some other random starting values for `n`:" ] }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "2.0" + "{42: 2.0000000000000315,\n", + " 99: 2.000000000000074,\n", + " 100: 2.0,\n", + " 212: 2.000000000000317,\n", + " 333: 2.0,\n", + " 538: 2.0000000000004023}" ] }, - "execution_count": 16, + "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "n = 250; diff(n, 2*n) / diff(2*n, 4*n)" - ] - }, - { - "cell_type": "code", - "execution_count": 17, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "2.0" - ] - }, - "execution_count": 17, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "n = 500; diff(n, 2*n) / diff(2*n, 4*n)" + "{n: diff(n, 2*n) / diff(2*n, 4*n) for n in (42, 99, 100, 212, 333, 538)}" ] }, { @@ -603,17 +456,16 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "array([[ 4.70916321e-33, -1.30373854e-30],\n", - " [ -1.30373854e-30, 1.44885627e-27]])" + "(0.43233235838169376, 0.2969970751450662)" ] }, - "execution_count": 18, + "execution_count": 13, "metadata": {}, "output_type": "execute_result" } @@ -621,39 +473,14 @@ "source": [ "from scipy.optimize import curve_fit\n", "\n", - "Ns = list(range(100, 5001, 100))\n", + "def estimated_density(n, A, B) -> float: return A + B / n\n", "\n", - "def f(n, A, B): return A + B / n\n", + "xdata = list(range(50, 5001, 100))\n", "\n", - "((A, B), covariance) = curve_fit(f=f, xdata=Ns, ydata=[density(n) for n in Ns])\n", + "result = curve_fit(xdata=xdata, ydata=[density(n) for n in xdata], f=estimated_density)\n", + "\n", + "A, B = map(float, result[0])\n", "\n", - "covariance" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The `curve_fit` function returns a sequence of parameter values, and a covariance matrix. The fact that all the numbers in the covariance matrix are really small indicates that the parameters are a really good fit. Here are the parameters, `A` and `B`:" - ] - }, - { - "cell_type": "code", - "execution_count": 19, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "(0.43233235838169332, 0.29699707514503415)" - ] - }, - "execution_count": 19, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ "A, B" ] }, @@ -661,16 +488,16 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "We can plug them into a function that estimates the density:" + "We can accept these A and B values:" ] }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 14, "metadata": {}, "outputs": [], "source": [ - "def estimated_density(n): return A + B / n" + "def estimated_density(n, A=A, B=B) -> float: return A + B / n" ] }, { @@ -682,34 +509,34 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "6.106226635438361e-16" + "1.6653345369377348e-16" ] }, - "execution_count": 21, + "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "max(abs(density(n) - estimated_density(n))\n", - " for n in range(200, 4000))" + "float(max(abs(density(n) - estimated_density(n))\n", + " for n in range(200, 2000)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "That says that, for all values of `n` from 200 to 4,000, `density(n)` and `estimated_density(n)` agree at least through the first 15 decimal places!\n", + "That says that, for all values of `n` from 200 to 2,000, `density(n)` and `estimated_density(n)` agree at least through the first 15 decimal places!\n", "\n", "We now have a plausible answer to the puzzle:\n", "\n", - "> lim n→∞ density(n) ≅ A = 0.43233235838169" + "lim n→∞ density(n) ≅ A ≅ 0.43233235838169" ] }, { @@ -718,28 +545,28 @@ "source": [ "# Why?\n", "\n", - "Theis answer is empirically strong (15 decimal places of accuracy) but theoretically weak: we don't have a **proof**, and we don't have an explanation for **why** the density function has this form. We need some more mathematical thinking. \n", + "This answer is empirically strong (15 decimal places of accuracy) but theoretically weak: we don't have a **proof**, and we don't have an explanation for **why** the density function has this form. We need some more mathematical thinking. \n", "\n", "I didn't have any ideas, so I looked to see if anyone else had written something about the number 0.432332358169 I tried several searches and found two interesting formulas:\n", "\n", "- Search: [`[0.4323323]`](https://www.google.com/search?q=0.4323323) Formula: `sinh(1) / exp(1)` [(Page)](http://arxiv.org/pdf/1310.4360.pdf)\n", - "- Search: [`[0.432332358]`](https://www.google.com/search?q=0.432332358) Formula: `0.5(1-e^(-2))` [(Page)](http://www.actuarialoutpost.com/actuarial_discussion_forum/archive/index.php/t-52095.html)\n", + "- Search: [`[0.432332358]`](https://www.google.com/search?q=0.432332358) Formula: `0.5(1-e^(-2))` [(Page)](https://math.stackexchange.com/questions/1330391/how-to-calculate-e-sin2x)\n", "\n", "I can verify that the two formulas are equivalent, and that they are indeed equal to `A` to at least 15 decimal places:" ] }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "(0.43233235838169365, 0.43233235838169365, 0.43233235838169332)" + "(0.43233235838169376, 0.43233235838169365, 0.43233235838169365)" ] }, - "execution_count": 22, + "execution_count": 16, "metadata": {}, "output_type": "execute_result" } @@ -747,10 +574,7 @@ "source": [ "from math import sinh, exp, e\n", "\n", - "S = sinh(1) / exp(1)\n", - "E = 0.5 * (1 - e ** (-2))\n", - "\n", - "S, E, A" + "A, sinh(1) / exp(1), 0.5 * (1 - e ** (-2))" ] }, { @@ -767,10 +591,13 @@ { "cell_type": "markdown", "metadata": { - "collapsed": true + "collapsed": true, + "jupyter": { + "outputs_hidden": true + } }, "source": [ - "# John Lamping to the Rescue\n", + "\n", "\n", "I reported my results to Anne Paulson and John Lamping, who had originally related the problem to me, and the next day John wrote back with the following:\n", "\n", @@ -835,15 +662,54 @@ "source": [ "# Validation by Anticipating Objections\n", "\n", - "Is our implementation of `occ(n)` correct? I think it is, but I can anticipate some objections and answer them:\n", + "Is our implementation of `occupancy(n)` correct? I think it is, but I can anticipate some objections and answer them:\n", "\n", - "- *In `occ(n)`, is it ok to start from all empty houses, rather than considering layouts of partially-occupied houses?* Yes, because the problem states that initially all houses are empty, and each choice of a house breaks the street up into runs of acceptable houses, flanked by unacceptable houses. If we get the computation right for a run of `n` acceptable houses, then we can get the whole answer right. A key point is that the chosen first house breaks the row of houses into 2 runs of *acceptable* houses, not 2 runs of *unoccupied* houses. If it were unoccupied houses, then we would have to also keep track of whether there were occupied houses to the right and/or left of the runs. By considering runs of acceptable houses, eveything is clean and simple.\n", + "- *In `occupancy(n)`, is it ok to start from all empty houses, rather than considering layouts of partially-occupied houses?* Yes, because the problem states that initially all houses are empty, and each choice of a house breaks the street up into runs of acceptable houses, flanked by unacceptable houses. If we get the computation right for a run of `n` acceptable houses, then we can get the whole answer right. A key point is that the chosen first house breaks the row of houses into 2 runs of *acceptable* houses, not 2 runs of *unoccupied* houses. If it were unoccupied houses, then we would have to also keep track of whether there were occupied houses to the right and/or left of the runs. By considering runs of acceptable houses, eveything is clean and simple.\n", "\n", - "- *In `occ(7)`, if the first house chosen is 2, that breaks the street up into runs of 1 and 3 acceptable houses. There is only one way to occupy the 1 house, but there are several ways to occupy the 3 houses. Shouldn't the average give more weight to the 3 houses, since there are more possibilities there?* No. We are calculating occupancy, and there is a specific number (5/3) which is the expected occupancy of 3 houses; it doesn't matter if there is one combination or a million combinations that contribute to that expected value, all that matters is what the expected value is.\n", + "- *In `occupancy(7)`, if the first house chosen is 2, that breaks the street up into runs of 1 and 3 acceptable houses. There is only one way to occupy the 1 house, but there are several ways to occupy the 3 houses. Shouldn't the average give more weight to the 3 houses, since there are more possibilities there?* No. We are calculating occupancy, and there is a specific number (5/3) which is the expected occupancy of 3 houses; it doesn't matter if there is one combination or a million combinations that contribute to that expected value, all that matters is what the expected value is.\n", "\n", "\n" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Validation by Unit Tests\n", + "\n", + "Another way to gain more confidence in the code is to run a test suite:" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'ok'" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def test():\n", + " assert occupancy(0) == 0\n", + " assert occupancy(1) == occupancy(2) == 1\n", + " assert occupancy(3) == 5/3\n", + " assert density(3) == occupancy(3) / 3\n", + " assert density(100) == occupancy(100) / 100\n", + " assert possible_runs(3) == [(0, 1), (0, 0), (1, 0)]\n", + " assert possible_runs(7) == [(0, 5), (0, 4), (1, 3), (2, 2), (3, 1), (4, 0), (5, 0)] \n", + " return 'ok'\n", + "\n", + "test()" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -856,28 +722,37 @@ "- When two very different implementations get the same result, that is evidence supporting both of them.\n", "\n", "\n", - "The simulation will start with an empty set of occupied houses. We then go through the houses in random order, occupying just the ones that have no neighbor. (Note that `random.sample(range(n), n)` gives a random ordering of `range(n)`.)" + "The simulation will start with an empty set of occupied houses. We then go through the houses in random order (order of attractiveness as John Lamping puts it), occupying just the ones that have no neighbor." ] }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "import random\n", "\n", - "def simulate(n):\n", - " \"Simulate moving in to houses, and return a sorted tuple of occupied houses.\"\n", + "def simulate_houses(n: int) -> set[int]:\n", + " \"Simulate moving in to houses, and return a set of occupied houses.\"\n", " occupied = set()\n", - " for house in random.sample(range(n), n):\n", + " for house in random_order(range(n)):\n", " if (house - 1) not in occupied and (house + 1) not in occupied:\n", " occupied.add(house)\n", - " return sorted(occupied)\n", + " return occupied\n", "\n", - "def simulated_density(n, repeat=10000):\n", + "def random_order(items): return random.sample(items, len(items))" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [], + "source": [ + "def simulated_density(n, repeat=10_000):\n", " \"Estimate density by simulation, repeated `repeat` times.\"\n", - " return mean(len(simulate(n)) / n \n", + " return mean(len(simulate_houses(n)) / n \n", " for _ in range(repeat))" ] }, @@ -885,89 +760,62 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Let's see if the simulation returns results that match the actual `density` function and the `estimated_density` function:" + "Let's measure the deltas for the simulation results (and the `estimated_sensity` results) compared to the actual `density` results:" ] }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 20, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - " n simulate density estimated\n", - " 10 0.4621 0.4620 0.4620\n", - " 25 0.4442 0.4442 0.4442\n", - " 50 0.4383 0.4383 0.4383\n", - " 100 0.4354 0.4353 0.4353\n", - " 200 0.4339 0.4338 0.4338\n", - " 400 0.4331 0.4331 0.4331\n" + " n density Δsimulated Δestimated\n", + " 50 0.4383 -1.7e-04 +1.7e-16\n", + " 100 0.4353 -5.5e-05 +5.6e-17\n", + " 150 0.4343 +3.2e-05 +5.6e-17\n", + " 200 0.4338 -2.7e-06 +0.0e+00\n", + " 250 0.4335 -9.8e-05 +0.0e+00\n", + " 300 0.4333 -2.7e-05 -5.6e-17\n" ] } ], "source": [ - "print(' n simulate density estimated')\n", - "for n in (10, 25, 50, 100, 200, 400):\n", - " print('{:4d} {:.4f} {:.4f} {:.4f}'\n", - " .format(n, simulated_density(n), density(n), estimated_density(n)))" + "print(' n density Δsimulated Δestimated')\n", + "for n in range(50, 301, 50):\n", + " s, d, e = simulated_density(n), density(n), estimated_density(n)\n", + " print(f'{n:4d} {d:.4f} {d-s:+9.1e} {d-e:+9.1e}')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "We get good agreement (at least to 3 decimal places), suggesting that either our three implementations are all correct, or we've made mistakes in all three. \n", + "We get good agreement, to at least to 4 decimal places for the simulated runs and 16 decimal places for the estimated density, suggesting that either our three implementations are all correct, or we've made similar mistakes in all three. \n", "\n", "The `simulate` function can also give us insights when we look at the results it produces:" ] }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "[0, 2, 4, 6]" + "Counter({(0, 2, 4, 6): 3264,\n", + " (1, 3, 5): 1431,\n", + " (0, 2, 5): 1248,\n", + " (1, 4, 6): 1192,\n", + " (1, 3, 6): 1036,\n", + " (0, 3, 5): 981,\n", + " (0, 3, 6): 848})" ] }, - "execution_count": 25, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "simulate(7)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let's repeat that multiple times, and store the results in a `Counter`, which tracks how many times it has seen each result:" - ] - }, - { - "cell_type": "code", - "execution_count": 26, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "Counter({(0, 2, 4, 6): 3220,\n", - " (0, 2, 5): 1214,\n", - " (0, 3, 5): 1029,\n", - " (0, 3, 6): 827,\n", - " (1, 3, 5): 1417,\n", - " (1, 3, 6): 976,\n", - " (1, 4, 6): 1317})" - ] - }, - "execution_count": 26, + "execution_count": 21, "metadata": {}, "output_type": "execute_result" } @@ -975,7 +823,7 @@ "source": [ "from collections import Counter\n", "\n", - "Counter(tuple(simulate(7)) for _ in range(10000))" + "Counter(tuple(simulate_houses(7)) for _ in range(10_000))" ] }, { @@ -985,58 +833,6 @@ "That says that about 1/3 of the time, things work out so that the 4 even-numbered houses are occupied. But if anybody ever chooses an odd-numbered house, then we are destined to have 3 houses occupied (in one of 6 different ways, of which (1, 3 5) is the most common, probably because it is the only one that has three chances of getting started with an odd-numbered house)." ] }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Verification by Test\n", - "\n", - "Another way to gain more confidence in the code is to run a test suite:" - ] - }, - { - "cell_type": "code", - "execution_count": 27, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "'ok'" - ] - }, - "execution_count": 27, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "def test():\n", - " assert occ(0) == 0\n", - " assert occ(1) == occ(2) == 1\n", - " assert occ(3) == 5/3\n", - " assert density(3) == occ(3) / 3\n", - " assert density(100) == occ(100) / 100\n", - " assert runs(3) == [(0, 1), (0, 0), (1, 0)]\n", - " assert runs(7) == [(0, 5), (0, 4), (1, 3), (2, 2), (3, 1), (4, 0), (5, 0)]\n", - " for n in (3, 7, 10, 20, 100, 101, 200, 201):\n", - " for repeat in range(500):\n", - " assert_valid(simulate(n), n) \n", - " return 'ok'\n", - "\n", - "def assert_valid(occupied, n):\n", - " \"\"\"Assert that, in this collection of occupied houses, no house is adjacent to an\n", - " occupied house, and every unoccupied position is adjacent to an occupied house.\"\"\"\n", - " occupied = set(occupied) # coerce to set\n", - " for house in range(n):\n", - " if house in occupied:\n", - " assert not ((house - 1) in occupied or (house + 1) in occupied)\n", - " else:\n", - " assert (house - 1) in occupied or (house + 1) in occupied\n", - "\n", - "test()" - ] - }, { "cell_type": "markdown", "metadata": {}, @@ -1049,13 +845,147 @@ "\n", "I got the fun of working on the puzzle, the satisfaction of seeing John Lamping work out a proof, and the awe of seeing the mathematically sophisticated solutions of Jim Ferry and Andrew Mascioli, solutions that I know I could never come up with, but that I could get near to by coming at the problem with a different style of thinking." ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Postscript: Another Puzzle Solved by Simulation\n", + "\n", + "The [538 Riddler column from Jan 28, 2021](https://fivethirtyeight.com/features/can-you-cut-the-square-into-more-squares/) poses this question:\n", + "\n", + "> *Robin of Foxley has entered the FiveThirtyEight archery tournament. Her aim is excellent (relatively speaking), as she is guaranteed to hit the circular target, which has no subdivisions — it’s just one big circle. However, her arrows are equally likely to hit each location within the target.*\n", + ">\n", + ">*Her true love, Marian, has issued a challenge. Robin must fire as many arrows as she can, such that each arrow is closer to the center of the target than the previous arrow. For example, if Robin fires three arrows, each closer to the center than the previous, but the fourth arrow is farther than the third, then she is done with the challenge and her score is four.*\n", + ">\n", + ">*On average, what score can Robin expect to achieve in this archery challenge?*\n", + "\n", + "The tournament can potentially have an infinite number of arrows, so the mathematical solution will be the sum of an infinite series. I don't have a good idea how to figure that out, but I know I can do a simulation. I'll make the target be the unit circle, and represent the point an arrow hits by a complex number, which is a point in the (real, imag) plane. Then the distance to the center of the target is just the absolute value of the point." + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [], + "source": [ + "Point = complex\n", + "distance = abs\n", + "\n", + "def random_arrow() -> Point:\n", + " \"\"\"A random point within the unit circle.\n", + " Found by sampling uniformly from a bounding square until we get a point inside the unit circle.\"\"\"\n", + " while distance(arrow := Point(random.uniform(-1, 1), random.uniform(-1, 1))) > 1:\n", + " pass\n", + " return arrow" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's make sure the random arrows look right:" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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PP8fgwYNx5swZXHLJJfjpT3+K0tJS5jFOnTqFU6dORf5ua2uTnAlBmONliX1RzIR4stS6cRLDZDZHO6n0ZpYVp+vJs1xo8++RkS5Uo0Z0vczcateUDJAau1uY3a+17x0lyw7hGE8VnGPHjuHMmTPo379/1Of9+/dHS0uL5f7BYBCbNm3C2rVroz4vLi7G008/jZEjR6KtrQ2PPPIIJk6ciN27d+Piiy+OOc6SJUvw4IMPOpsMQZigvdXr30g1zqgqdhxsxTUl3ioVVkI8nhkwVoKX9b1e6CsAFl5WhPmTiqQbcLK2j1yf9XvRibDpmmd1EVUa7K4nz3JxvP00lm7aF3X9zIS7jHvHbbeaG/DGlJ2RlvJB8UR8iEuQsaJER+ypqhrzGYunn34avXv3xowZM6I+Hz9+PL797W+jpKQEl112Gf7whz9g6NCheOyxx5jHqaqqQigUivw7fPiw7bkQBI/ZYwdh46IJYN3ZdzzX6Lmryi+9gKyCeVnfG4W+CuCpbU3uB2srhv9Kjt0NeJaLaoNyA/Br1PBivl7YfYTp0vHLvSEyphMdZ1I+KJ6ID55acHJzc5Genh5jrfn4449jrDpGVFXFb37zG8ybNw8ZGRmm26alpWHs2LF4//33md9nZmYiMzNTbvAEYQOtxL5mKdCI11tool1RVsG8vO8fnlPC7qzu0rrtPtwa1fOLddx4pdOzLBdpiHVvapxRVew81IqcHuesSryYr9vX7eJacxJ9b7BgjSkYak/5oHgiPnhqwcnIyEBZWRm2bNkS9fmWLVswYcIE031fffVV/OMf/8CCBQssz6OqKnbt2oX8/HxH4yUIN5g9dhAenRsbDxavt9B4dtc2YhXMy/s+TVG4FXedrltNQ3O435OFVSBe6fQsy8W9FcXc+SsKcNvaxiirklWFYl5l4kTeGzyMY/KjtYlITjxPE7/rrrswb948jBkzBuXl5XjqqafQ3NyMW2+9FUDYfXTkyBH87ne/i9pv9erVGDduHEaMGBFzzAcffBDjx4/HxRdfjLa2Njz66KPYtWsXnnjiCa+nQxBClA3O6ZJvoVZxQLzvRw/OiYqR0eNk3XjZQ0A4buXY518gGGpHfiDLlZ5eZuPQx/WwLBe9s7vFzD9NAVQ1tl9VXeWUqFR2IzLNVuOJaHyTH61NRPLhuYIze/ZsfPLJJ/jJT36CYDCIESNG4KWXXopkRQWDQTQ3R/u5Q6EQ1q9fj0ceeYR5zOPHj+Pmm29GS0sLAoEASktLUVtbi0svvdTr6RCEEF21OaHVvM2+14TamrqDWFX3ATrVc2/vAFB/4JhQdpVeiPKyhxSEFQejS8eLa7ai9gCqN+2DyggGVo2qlwJADVttFk66CCMv7IXb1+2K2sSYGbfzUCtuW9sYdSQ/KtOyNW+oLQThFEVVGep/itPW1oZAIIBQKIRevXolejhEChMMtSfFW6jbLSas5i3zvUzKsFGI3jutGEs3RwfvntUhokhXFNRVTonEgLh1zVa8egBLNu2L+ixNARZdMQTLXzlgOs50RcGGReXMflXaWPXzNipm8UirFr1vgqF2TKzeajkPgrBCRn6TgkMKDpHEuKGYiLxZJ6rHloxg5G27uGIYlm3aHxH+35tUiJXbmmLOtW7heJQP6csdh+z8g6F2TFiylekeM5IGxLjmtDE1f3pCSHmJtzItY5GpP3AMc1e+EfO52ZoTBAsZ+U3NNgkiDsgISNFt3ShzL5I5lEgFyCzw13ge3rajBvZGXeWUiPAHgNV1TcKxNnbXuenYCSHlBggrN8rZeBvjmMqH9BWKRzFrtur2tZHNOPMyvokgeJCCQxAeIyMgRbd1K6XZSoFwSwGyi4xgNNvWKPxFY22crDOvujULlqXJGLtkRznx6trIKJ5A141JIxILKTgE4QCRir2iAlJmW56AeXFPENNH5QsLDisFwqkC5NR6ICMYZbYVzdKRFeRm4+GhdztdWzLANTeTl3V97FhkKDOKiDek4BCETUTejt1wsbC25VkHfvbiu/jFS+8Kv6nb7ZwtogC51U9IRjDKbCtiFXHqWtGPZ8+Hx7Fs8zkLzeJpwzDqwt5R43Qzc8iJcmaFXYsMZUYR8YQUHIKwgejbsVsuFiNm1gHZN3UzpSA/kBXO8Nm0D52ILboWr35CMoLRTSFa+97RqLgYReH3sLIaT/mQvrj2EvcsNFbYUc5kLG5kkSH8Tlx6URFEqiFa9VamKqtsBdfZYwehrnIKfjR9eMx3shV4eRVuaxqaw+nLCAv3xRXDoqwwqdxPiFkkUAWyM9KZVYJFiGclYdn7yU4fLj9WRiYIDbLgEIQNZN6O7bpYsjPScKLjTKTSrh79m/b0Ufn4xUvvupahoh27R0Z6dANMFVi2aT+uLRkQNZ5U6SdktF6wlFgV5v2e/Ibovcds3rl+L4rzeqKkwP2O44kqO0B0LUjBIQgbyMYgyLpYzOJXWLE/bmWo6I/NKojHi+Ewzi/ZsmZYazp5aD9uFlS8mqe6gci9x2zeCWDG8npUn7333FJK7GZ2kVJEyEKF/qjQH+EAs+Jqdh/IrAJxWnE7ANzCdwBsx0MEQ+3YcagVtxtK/huRrT6bDJWczYoJ1r531DQLKlUK1bHWQENLYV+6aZ/jgHG7FY29LEVAJBdU6I8g4gTv7djJA/k3dU1cy4kKlRvbYjcWQj9WFpoVw44VJhmyZsziqdzu9+RXK4RmcWM1Oz2jqpFeWoAz65WdzC4v092J1IYUHIKwgZmgchLPEAy1Y3VdbBuBNCAiTN2MbTGO1YjWD+lkR6evrTAi8K6ZVTxVfiAL00dl4fNTXzpyufndCjF77CAU5/XEjOX1UZljaYh109lNN7eT2eVlujuR2lAWFUFIYpVtYhbPYJWZwut+fdPkoog1RCYzxgre+fTHLinISfpMGdY1C4baUX/gGAAIramWtbZu4XjUVU6RUk54Vgi72VheUVKQg2rDWlw1Ii9mO7tKtZ37V1OKjOfPzkhD/YFjvltDwj+QBYcgJBAxl/OK8KkCpnXWvmkKMH9iUeRvu/VHWBYM3vkenVOKssIc20qNF7237MK6ZpUb9gJqOIhas6bo+1W5ESyuJ5msEMZMvpnL62O2WVwxzPa4Ze9fVsD6jNIBkS7rfrSGEf6AFByCkEBEUFnFM5gJNdHsI1lBy3OP8M53TckA4WOLnsvptkZEFSNmurfub01JraucYhow7EQR82uzSd6ctPur/sAxpoVv1MDepvtbIXv/spQuiskhrCAFhyAkEBVUvHgGEaHmdoVYK6uTm+dzo/dWcV5PnOg4Yyo0ZRQjkaaXVoqn0/gZP6bNi8ypR0Z6TLkA7R6Od0yRmdLlV2sYkVgoBocgJJCJIWDFM8gINdU0YVsckarLblWkFa3wbLbtjCfqTavpysazaO0mzDBTPJ3Ez2hxPsFQe1QMz4ZF5Sjok52w+BGROdU0NGPm8voY5eYX140AAGYg/Z/3fOT5nHgxOYm2hhH+gyw4BCGJV80fNdx+M46ne8Rp7y3gnLVAE5pG64+deJaRFwa4Y7ZSPGXPp7lt9h4JMWvHuNWEVBa9O0m2SzwQHuuGReUoKchhWlE6Ady2ttHzOfnRGkb4E1JwCMIGspWJZQKB3a75EU+BIHMu47YsZacTwJq6g7hP12/LjsLmJJha5ny8mkJ691siaroYleZ7K4qlu8R3qsDJjnBUmZnbjzUntwPJqdEnIQIpOARhQrwLs3mVbeNUIHjVZdoYPDrjifoYx9yqug8wf1JhTBC3UYkCgD/v+QiqqmJMYZ8Y4XpvRTGWbdovHUwtqrRZ1RQ6o6poONga9/gRltK8dNO+qC7xaYBQl3h9bSBeN3vjnLyK1UmGIpJEYiEFhyA4JKIwm5fuJLsCgbcOZkqPXQvXwsuK8NS26EKHnSpiFACjElX73tGo9hYKgOpZIwEg2nIxrRijLuwtreRNHtoPj3zrEkAF1+Lz1sFPTQOZ0xUFYwtz4p5NxbPGvPaPY+cajhliWkSUOu0a7DjYijuea2TOyQuLpF+rQRP+gxQcgmDg9MHsJH1W1Frg5kOedzzeOhw/eRpLN5+LL1kwqQjfm1TkeCzzJxVhVV2TkAKgKUbaGPUyXAVQuX4vFJ0y0akCyzbvl+qlBYgputo2PPRFE+MdP8JzJ9W+fyzy/6z7W8QSlx/IwjUlWTgSao/EG+nn5HbGk9+rQRP+ghQcgmDgxFXk9CFsJVjcfsibHY+3DtWb9kUFA6/c1oRV25pQPUtuLEbFyk68EK8as4roejfa2K36HunHI6Lo8lxTaQjHuhgtRl6UATBTdvMDWVgwqQgrDZYxI6y1EbHE1TQ0R5QbBcDiacMi94CbFknqSUXIQgoOQTCw+2B26yHMEyxuP+StjscLzuUpFDJj0StWigJUVhTjlslDpBUAs4BXXg0Xq/FoVqmSgt6Wii5PwXpsbimmj2LH+LgVPyKq7H5vUhFWbYtt4qrHbvNQ/f2jImwlu/aSAbYVVh7JVA2a8AdUB4cgGMjUu9EjUwfGDm4f3+p4rHXQMnBYiI4lRjCqwJKX9mHFqwci5xWty6ONUTGMiaXc8K4hS9Fbua0Jt61tNIanxCgCvLosowfHNlbV18Vximx9noWXFUUe+OmKglmjBzruaSZyPzrp4aWH6t8QspAFhyA4aNWIGw62YmxhDrcTuN5FIGL58VPJf5HjsSwqvbO6Md0yomPhWT2WbtoXefuXQRvjzkOt+PREB/7rj3+PUm7ScK6Gi4ZVXRg92hqxFAFRK4XbrkVRi4bxvDdPuiiSlXbPVcMcucpE70c3LFZU/4aQhRQcguAgE1yq38bsIey3kv+ixzMKKE2hWPNaE1bVNqETclaAotweUJTYGJlOxGZMycxl+qgs/PzFd2JcMZ04V8MFiL0O/3HFkBiLj4YK4LE5peh7fiZXEbByq3kRPyKqTBvPu7quCfMnFQKIDtSuP3DMs6B4t6D6N4QMiqoyihikOG1tbQgEAgiFQujVq1eih0MkCDNLSjDUHpV2DISFhz4DJxhqx8TqrTECpq5yCgDEPIRZ26cBeHRuKcoGy3XuDobaXX3IOzme3X1X1B7Akpf2RX1mXGPt+DKdyY1rDITX+bWqr0WEOWsbHqwxyVJ/4Bjmrnwj5vN1C8ebNvi0oqahOUa50CvMIud1w7Jk9x6glG9CFhn5TRYcokti9VD/TV1sQKZIcKm2DSt+hFmPBPbK27td5MzJ8az25QmxWyYPAc4WneNZgGSFL8/VdNPkItPrxsNYAE8Wbe49MtI9qX9jZdGwsvJ4HRRvBqV8E15DCg7R5bB6qAdD7VhdF5tSmwYwg0tFhZZseftUwEqI3XL5EFx7yQCmgLYjfHlZX/MnFpluwyJNATYumsCNvbLCOPeZpQPxfONHrrtyzJQLKxeSk8wkJ9YXSvkm4gEpOESXw+qhLmIFAOTjD7Ttq9bvRSfje7+nvMoKNJ4QK87riRMdZ6Jq37COZ0f4ilwT4zYs0hBWxozKjegasOa+sfEIVn6nDNkZ3SLKnFMlQWRfMytPj4x0qVR6DafWF0r5JuJBXBSc5cuX47//+78RDAbx1a9+FQ8//DAuu+wy5ravvPIKpkyZEvP5u+++i+Li4sjf69evx/33348DBw5gyJAh+PnPf46ZM2d6NgcidbCyvIhYATRkgx5njx2EHpnn4ba1jTHfGS1EfsKOQOMJsRnL66EKHMduxpjINdFvs+fD41i2eX+k4edNuiwju2vAa49w0293oHrWSJQP6St0PJYSEwy1Y01dE1aerWsjcj1YSqR2ftFUev2YnFpf4tnhHqBYn66K5wpOTU0N7rzzTixfvhwTJ07EihUrUFFRgXfeeQeDBvF/kPv3748KIOrXr1/k/7dv347Zs2fjpz/9KWbOnImNGzfi+uuvR11dHcaNG+fpfIj4sPtwK948+CkuLezDdBE4eWBZveXbsczwvmONs2xwbD8iIFz11o8PX7sCjecKUgWPY7wOaQAWnM3+sUIkJkTbpnxIX66bTEN2Dbhzh3hXcZYCBCAmPd+OgrH7cCsqN+yNymJLU2JT6Vm4YX2JZ/YVxfp0XTzPoho3bhxGjx6NX//615HPhg8fjhkzZmDJkiUx22sWnNbWVvTu3Zt5zNmzZ6OtrQ2bNm2KfDZt2jTk5ORg3bp1lmOiLCp/c/cfdmH9ziORv2eNHoiHrr8k8rdbDyyrzA/99wCkFSqzceqzX9KUsHJzy+Qh0nNwE57SaJWJY6ZsRs0TYLrmrDKJwhaLg1hV90HChJSdLCj99Tfyo+nD8bMX3+Uej5dxB5PYIdGMrJqGZlSu38tMiRc5hln2YLwy8GSO79ZYCX/gmyyqjo4O7NixA5WVlVGfT506FfX19ab7lpaW4osvvsBXvvIV/OhHP4pyW23fvh0//OEPo7a/6qqr8PDDD7s2diIx7D7cGqXcAMD6nUfwnfLBKCnIcTU40eotX/vejkJlNU6/1fMw6xj+6YkObpyG1dro55mdkYaZy+tjXH8ibglNuQESE5Bqx6WiFYqc8UR9zNpZdRXnZdzxei2IundYjUllj+Gm9cXtbEAjFOvTtfFUwTl27BjOnDmD/v37R33ev39/tLS0MPfJz8/HU089hbKyMpw6dQq///3vceWVV+KVV17B5MmTAQAtLS1Sxzx16hROnToV+butrc3JtAgPefPgp8zP3zrYipKCHKkHlj5F90THmch/ZawwdhUqkXF6/XAXhdsxvP10VBNFTcnRBBqAmP2q1u+NWRv9PJdcNzLKNaKqQO17R6VTv+MtpOwK9ZKCHFTPit3Pqqs4Mw4MYFpwZFLZeQH0aYpcOrzfFHQePTLSmZ9nZ1CXoq5AXIKMFUOTGFVVYz7TGDZsGIYNGxb5u7y8HIcPH8b//M//RBQc2WMuWbIEDz74oN3hE3Hk0sI+zM/HFIbjAopyewhlfZi5B2RcHHaFKy9Q+djnXyAYandNILgRPGnaMVxTRBAe/2NzSlFWGC5KWH/gGNPKsKbuIO6bPpx5rslD+0VdPC0mRTb120lHarvrZVeo8/YzOx5PoQIQ5drkBUTz4ClOdtLh/aKgm3Gi4wzzc31VayJ18VTByc3NRXp6eoxl5eOPP46xwJgxfvx4PPPMM5G/8/LypI5ZVVWFu+66K/J3W1sbCgoKhM9PxI+SghzMGj0wJgZHe/jWvnc0ansFsW+eRouEkU41bHkQcXHYFa5GAaW1Jbh93S7XYkjcikXiCT1WFlDf8zOjrAys9gar6j7gCt0dh1otCygaccsl4sZ62RXqvP3MjsdTgJxYTnhrabfWTzzwU+82Irnw1E6XkZGBsrIybNmyJerzLVu2YMKECcLHaWxsRH5+fuTv8vLymGP+5S9/4R4zMzMTvXr1ivpH+JeHrr8Ef/z+BNw/fTj++P0JkQBjVvyAopy1CugQqVTbqQJrXmuyHIsmEOx0Xda6KD8xtxRQzykCmgvISUdp2U7SZrDmyOoYbhQM+YEsLLwsNnW+UwWzo3hNQzNuZ6THi6Z+O+lI7eZ6xRMty8to3RHttM7Cre7e8aCmoRkTq7di7so3MLF6K2oamqX2d/L7JZIfz11Ud911F+bNm4cxY8agvLwcTz31FJqbm3HrrbcCCFtXjhw5gt/97ncAgIcffhiFhYX46le/io6ODjzzzDNYv3491q9fHznmD37wA0yePBlLly7FN7/5Tfzxj3/EX//6V9TV1Xk9HSJOlBTEdu/m1RYxvv2LVqpdVduE+ROLLB92TuIN8gNZyOlxQtpqYYXbcSnMjuHZ3SytJvMnFWFVXZPlGzIvuFUm9sOJS8QPcTx+IhncS24lFGjB3g0HWzG2MPa5QqQunis4s2fPxieffIKf/OQnCAaDGDFiBF566SUMHjwYABAMBtHcfE4r7+jowD333IMjR44gKysLX/3qV/Hiiy/i6quvjmwzYcIEPPfcc/jRj36E+++/H0OGDEFNTQ3VwElxzMzNRjP2gklFWLnN3EIj0rlaf1y7TRGdmMl55nkvTO9GoSei2Im6j3hWtUfnlOKakgG2xywKuSqSD7eUUqqD03WhbuLkrkoqWN2TAcQ8wCYP7RfTDdyIVT0MNx+MVl2f7ZzfzjG9QqSmkFnndbOu7m5VoPXTeiUrXlcE1h8fgOMaNiJ1cKjKcXIhI79JwSEFJ+kwFuDjPcBq3zsaJdBmlA6IaXbIE3BeFAiTKWomen6vC6W5iahyqi+G6Pabt9PijV1ZGHptCeFVbhZRSu0WqSTrTvLhm0J/BOEFelcKK1VZM2OzXCz3XDVMSCHwImbD6AIyE5ai50+GWAoN4/UAopVTfYwFEFtjx43ifk6KN/pVGLqhdFkdw+vu37zj11VOQV3lFNPfrNl1sXJrU0fz1IYUHCKpsYqtMCoAogqB1zEbVsIyVWNGRJVTFapnQcF2BJtfhaEbSpfIMbwO0jY7vlnGmNV1MYsRM7v/SMFJDaicI5HU2E0DDYba8ec9H+GF3UeYqcJeppeKpCx3hfRWTYnToylxZt85xUyYurmP17iR+i5yDH27Dj1uKtx2r7fIdeGlxXt5jxH+gCw4RNIjm8ZtbDaoAKieFfvW6lU5equHsuYqSJZy+HaxysCyU9xPxF1jxzrmR4uaG1YVq2PorTusdh1u3ZN2izmKXheW5datApKEfyEFh0gJRF1PrHosKtg9lPTHDYbaUX/gmCvBpbyH8p4Pj+OGVa/HuAq8zlZx+/gyxzZT4uworiLuGjuCzY/C0A2lSyZGhdWuw03sKPROr0uqv0R0dSiLirKouhS8rArgXGaFES+CS40ZRYsrhkUaW2rIZG3JKBVeBssmKhDXTtabnQw0v2WtuZH6zjuGVQaSn/DbdSG8g7KoCN+S6DRbXpXjNID55utVcKnxzdGJu0FGqfAyWDaRgbh21s9OBprfstbcsEDwjuFHtxyPeFyXRD+7CHlIwSHihh/SbDWTduWGvZFO2QqAJbNGMh9aXmaPGB/KdoSJrFLh5XwS2Q4hmYSx27gh3HkxKvdWFEcsi1665fyuPPjh2UXIQwoOERf8lGarvbHuPNQKVYVpPEG8BKedWAItE0xGqfByPnuPhGI+i5eS4ccYGbeJtxJQ09AcUW4UAIunDfNEqPtdefDTs4uQgxQcIi74rdlhfiAL00d5H8Qog4y7QS8UjJgpFV7NJxhqx9JN+2I+XzxtWNyubyoHjMZbCWAFGC/bvB/XXjLA1XVNBuXBb88uQhxScIi4kMwuhHgKThF3g1Eo6BFRWLyYD6+Z5qgLezs+tgx+i5Fxg0QoAfES6smgPCTzs6urQ4X+iLhgVbhOS8OWKVIWT/IDWaYVVeMJT5m4f/rwqEJmZrg9Hyqa5h2JKDLoxfVk/caT4b7pCkU3UxWy4BBxY/bYQSjO64mGg60YW5iDkoIcAP73wdvFq5gJ3hvl1aPyE/bQTVQMjN+DU0Uxm0ciLAhOr6dxPrzfeLLETqWy+zOVoTo4VAcnbrAecpOH9nO9a7cfiEfnZV79E56wjIcyEM96JMYquwsvK8L8SUWmdW/8qAyJ3Csrag/EZDPFq76Q7PU0zufeacVYutm8xhPVsSFEkZHfpOCQghMXeIXYHp5TgtvX7YrZPlHFxPRCEICUQNT27ZGRjpnL6z1X2lhCgScs/WYlc6pssO4ngD83u/P3WikSKVBoVOQqK4pxy+VD4jZGGVjzSQPQydjWjwUDzfDTOndlqNAf4Tt4cQRpiuKbAD6jIAHOlae3EohR+yqA8bXBi8BJY0AtLxi1OK+nrzJV3FC2eHFIrLnZDdKNh1JoFWQbDLVH9U1TAVRv3hfJZvKb4sqaTydifxN+i7Oxwm/rTIhBQcZEXOAFE44enOOLAD5WWqz2PLbq0hyzr2TqtlvwhGXDwVbfdMJ2owM2wL6fNIxzsxOk69Y4rbAKst1xqBXG20lVgZ2HWuM2Rhl486msKJb6jbMCkhOViODHdSbEIAsOERfMggkTEcBnNDfzLAIaZhYY3r6aaT5eShsvGHVsYY4rVjK7Jnr9fm6lBWv3U9X6vUz3x54jxyPuDztBuvFKXzb7XQRD7djf0sbc79MTHb5MsebNZ/bYQbi2ZIB0jSfNWgIgYRYUP64zIQYpOETcMFNk4lm/hBfszOpRpaHvsGwU8jwBumFROU52dMZNaeMJl5KCnChlIA2IUbislBejC66yohi3TB4Ss52RFa8eQPWmfRFX373Tik2VDTudyB/72z+w9s3mqO+WbtqHa0sGRO4r2UydeGYusX4XZoUcAeDHf/o77q0wX8tEoc1nx8FWQAHKBoezJe3UeOpUw4qNqsZaVN3qn2Z1v7HuhTQFyM4gB4jfoSBjCjLuUpgFdda+dzQiBBUAOBs3oAlEgP8W6UZXZ7cQCT6+t6IYIwcGUJTbA3/a9VGUEmJ8O+YF9FYZgl2NrKg9gCUvRVc31jqnL9u0P2at7MY58Lpe3zy5CPdd/RXTdTEjUdeUt95G0hUFi6cNw7LNsWuZaNy+liycBinLjFF/L2hQLE5ioCwqC0jB6brwHqDaw1IvBAFE/b9VtotfU11FBaaGcV68NUsD8FrV15hzDYbaMaF6KzMead3C8SjMzY5aK5FsIuPx9dluzIwqk/GJkohrKivkjWuZaHYfbsWMJ+qjYofSFOC1SutrwczCOvuioT+e06xE2fsNYM8rFUpaJBuURUUQHKxcD0Yzul7IW/nh/domwCq+yIhxXkW5PZiZYZ2AaVwSS7lJAyKCWL+fTJwD6817waQirNzWJDw+URJxTXkuEZaQZ61lIqlpaEblhr0xgdGdKrDmtSbMn1hk6hLiuRMBuFoMcMchfuA977gnOs7EzIticfwNKTiE5/ipfoTdyql+6Eejr7NzouOM0HoGQ+14YfdHUufRlBCN/EAWKiuKme4m3vxZ6wWEXWOsMYuuLy+jZcOicqza1sRUAJKNeAl5t9GuDc8n8FRtE1bWNlmWXuDF6rmViFDT0IzK9XtjPre6X/zwDCDkIAWH8BQ/1o+wk7WV6JLyrKBTkbgBfQ0VUYxKSDDUjpEDA/j+FUPw61cOCGWGGddLi/vhBSbHbA9gwaTCmO14lp6THZ2onuXt9Ymnou6VkPdyDiKWQmOgcHFeT6ayzrJKuWGpiihhhs/TlNjAeyOJfgYQ8lAMDsXgeIYdP3e8xmX3Ie8kJsNJmjUvhoa3nlZxN1oKuz6YmqWEsIKTRw3szZw/a36y6xUMtWNN3UGsqvuAqRRb3VNexcz4UVGXxekcrO5f2VgvPV6sKWu8vPimx79VimtKBggf108xT10NisEhfIEf60c4fcjbfYt0cl6zN2PeeprtY0xhB8B8YLPcQcs27WcqVGbNFGXXS1NutHPqU4JF3qJVaZuVOXYrIfsJp3MQuX+N10ZRAKjnqoEbY4j0uL2mvPEW5fbA2WFFSFcUlBXmCB/bTzFPhDmk4BCe4bXPWtYikihB5fS8ez8Mcb/jrScvBkYzxWud3DVECxiyFCo311XknDz3jVdWFj8q6rI4mYPM9TVeG+Cc8qwvw8DCrTXdfbg1HOhsGG9xXk/8eU8walsF1q4pInkhBYdwHb3i4ZXPmlesz0zhSZSgcipclm7ex/zObD1ZMS03TS7C/In8bttGeNk8xz7/AsFQe+Q4bq6rqFJsfIv2UnlNhUJvTl42ZK8vLxNRU35e3BPEz158N2Y/Y3C7DNozZ++HoUhNJ+N4v/lEfcx+ihKOayJSE1JwCFdhKR51lVNc9VmzhFnlhr1R5nDW27tXFiUrS5KT87518FOmq+n+6cNx9ah80/V02gKD5XJQVeD2dbtizP5uravdQE4vlVfjmIDwXGcur/dVLI7ZfegkQNbt6zt9VD5+8dK7MddrzqUF0scD2AH4onSq1qUE/JQFSshBQcYUZOwaokHFvAeG6INEpBAaL/jW7eq0om4RO+flPbi9DNTmBQrvPNSK29Y2coucub2udoKTvQ5o91OhN+N1Er0PjYUsRQW3F7+bqKrhOPdysmBSEb43qUhofCKBzcaYGz1W1y8VgstTDQoyJhKCyFs074Eh8yDhxZeYnVfDzcaeTmITZOMeNFh9pNzCLFA4p8cJ0yJnbjdMlQ3kjEcKr18KvbEy25Zu2id0H2rrKiu43b6+2vGMinOnCqzc1oSV25oiionZ+KxS09MArLyxDAt/t4Mbj2amPCV7cHlXJy5O5OXLl6OoqAjdu3dHWVkZtm3bxt12w4YN+MY3voF+/fqhV69eKC8vx//93/9FbfP0009DUZSYf1988YXXUyFM0BQPPWkAPjlxCsFQO/eBsftwK/PzYKideR5NmKUr4ZOlKYDhtKYm9PxAFsqH9HX8kDJT6Jyel/fgfmxuqSdvkLxro10D1rU1rrFb62qX2WMHoa5yCtYtHI+6yimur5PIGrhJMNSO+gPHon4HrOtUrVNuNMzuQ6trzcPt6xtWnDO41hW90sMbH+uaaKQrCpbMGokrh+dFPy8Q7lP2WuXXTO8R2d834T88t+DU1NTgzjvvxPLlyzFx4kSsWLECFRUVeOeddzBoUOzNVVtbi2984xv4xS9+gd69e2PNmjX413/9V7zxxhsoLS2NbNerVy/s378/at/u3bt7PR3ChJiYDYQfUretbUSaAtw0qYj5wGg4KF823fhGqc/QcOPt3W6XYbcEHu/YoweLp7PKYGV941lIgLDL0C/xCU5SeIOhdrx18FMoioKywTlM68fM0oFYv/NI5LMZpQPiak1jXSdWkIHZfeinrDARayzAHx/rvlxcMSymVpMdCxRVLk5+PI/BGTduHEaPHo1f//rXkc+GDx+OGTNmYMmSJULH+OpXv4rZs2fjv/7rvwCELTh33nknjh8/bmtMFIPjDfpWAoc/bccdzzVGZ54AAOOBsWFROWYur2d+LtOS4K2DnyLtrBLg5EFtt8uw292c49nNWiZ+Sq9UuhmfYFQq7ZQBkA0G5WXfKACqZ1l3VfciBsfsPAC7sShwrnij1b3itwKcIkHCVuMzi9lyEiScqI7yBB/fxOB0dHRgx44dqKysjPp86tSpqK+PTdlj0dnZic8++wx9+vSJ+vzzzz/H4MGDcebMGVxyySX46U9/GmXh0XPq1CmcOnUq8ndbW5vkTAgrjEoBy1rTCeDmSRdhdV1T1AOjpCAn5i1sRumAiNIjomS4JWhF/e7aQ3Py0H6uZ4lpuB33YIZoDItmIXE7PsF4DWeWDsTGxiPC19TOPWAmWFUAVRv2Rs0nXpYPs/OUD+nLbCwKhN2XfXpkMgs26gV8POKVZNDu8zWvNWFVbVO4wrauSKDo+FgFHp0+G6x+g5Rh5W88VXCOHTuGM2fOoH///lGf9+/fHy0tLULHeOihh3DixAlcf/31kc+Ki4vx9NNPY+TIkWhra8MjjzyCiRMnYvfu3bj44otjjrFkyRI8+OCDziZDcGEJu1Xbmpjm3fmTCjF/UmHMA0P/IMnOSIuy6JgJTytBK/sAchIozVsbJw9AJy4XWWQUKjeFPesa6t1AVsqTHWWLF8Stx5hCHC+XhdV5vjepiNlYlGW55N2r8VSeRcgPZOG+q7+C+ROLmEUCzcbHm6NbSjjvN0gZVv4nLkHGihIdBaaqasxnLNatW4cHHngANTU1uOCCCyKfjx8/Ht/+9rdRUlKCyy67DH/4wx8wdOhQPPbYY8zjVFVVIRQKRf4dPnzY2YSIKFjCrhPATZMuigT26d/CeMGK2ucnOs4IB/eZCdqahmZMrN6KuSvfwMTqrahpaLaci1UgqUyApp3zJxrRQFI3A25FmjSaBXfaCQYVOWeaEttVXR+s6pXlw+o8+YEsVM+yHofVvZrooHAW+jGJjM9sjl4GCdsN1Cbii6cWnNzcXKSnp8dYaz7++OMYq46RmpoaLFiwAP/7v/+Lr3/966bbpqWlYezYsXj//feZ32dmZiIzM1Nu8IQwvDdOnrXG7vFEWxKkKwqyM9Jsvb1Zme8T0b5ABLdN5VbHM65TGoDF04bZOrdIoKmZ8mTHsmJ1TuXsG7kbZQbsXJvJQ/vhkW9dAqhAWWGsZYY3Dv25/BRM7BVmc/TS4tYV1jYV8NSCk5GRgbKyMmzZsiXq8y1btmDChAnc/datW4fvfve7WLt2LaZPn255HlVVsWvXLuTn5zseMyGP2RunnbdEmTdl3rYyViAjZunGopaLeKaYWlmKWOnGTo6nMXvsICyuGAZFCVvslm7eZ8tKZbyGLBZX8JUnO5YV4z4KzpUaUABUVhRz3Q0y97QdK562z21rG3HHc42ofe+o0DiM59p7JBRbtkHXagOQvzf8htnvMaacBOwr4TLnJfyD51lUNTU1mDdvHp588kmUl5fjqaeewsqVK/H3v/8dgwcPRlVVFY4cOYLf/e53AMLKzXe+8x088sgjuO666yLHycrKQiAQAAA8+OCDGD9+PC6++GK0tbXh0Ucfxe9//3u89tpruPTSSy3HRFlU3iBbfdbN4xm39TJTRCSzwg8ZN/kB+YJuMuN2e47BUDu3T9G6heNRPqSv5f6y918w1I4dB1tjM/4U4JE5l2BMYR9Haeey62N3TXn7LZ42DMs272d295YN5PYrVr/HFbUHwhlyLs+TMqwSg2+yqABg9uzZ+OSTT/CTn/wEwWAQI0aMwEsvvYTBgwcDAILBIJqbz73VrFixAl9++SW+//3v4/vf/37k8xtvvBFPP/00AOD48eO4+eab0dLSgkAggNLSUtTW1gopN4R3uB0QK3M847ZeZoqIuCnilaliZSmSdZPJmN7dNtPnB9h9ikTfjEXuF1ZGUZ/zGTFkamzPLVnsrI/dNeXtN+rC3qirnMKsGCwTyO1nzH6PwVA7lp5VbgB35+m3QG0ilri0ali0aBEWLVrE/E5TWjReeeUVy+P96le/wq9+9SsXRkakMl4+gESEaTwegGZxBnaEpRvxT07M9F4qhjxrllk8jhOB2CMjnfm5WRdyu2tqtl9+gN1qw0g8Yki8Sqvm/R69jpWJZ5YjIU9csqgIIlEkOlPE6/PHxBkowPcmFQKwFyfgRvyTG2/Gbrdc2H24FZXr2VkvVjFAdmOnTnScYX5++FN+3IvdNbXaz6ylgYbXMSSJyCqkWJmuDXUTpxgcIsG48VYbDLVjzWtNWFnbFNWgEIBUnIC+GvXJjk5b8U9+o6ahGZUb9jJbGuhje0S6psvAiotREM7Qsop7sbumZvsZY0ZmlA7A840f+apSthdQrExqISO/ScEhBYdIIG4VC7Mq7++kYFoyw1oXDZ6AdUMgRlpAHAlh2ab9kVR6FZBWntx067CC8eOhnNYfOIa5K9+I+VwkeNwN/K6EE+L4KsiYIAg2btbKsSrvL2KFiUfdnniXtucV9EsDuK4fp7FTRkXx3mnFGHVhb3xy4hRuW9sYta1VPIjbSicrGD8e1yHRjSu9mie1avA3pOAQRIJwMwDSqQCJR+Eyu8LaiRBhrUuaAmxcNAElBfzO7HYFIktRXLZ5P+oqp6AwNzt2LDgXdMxqNhrPYpFeEq+sQq/RXyMnzWZJMYoPpOAQRIJw863WqQDx+g3brrB2asHgrYuZcqPnb++2YOu+j/G14gtw5fA8y+2tLGn6sQDhAokzl9cza9IU9Ml2Ren0izB1I6swkXPR34ta3LI+7V5U+UxFV7BfIQWHcBW/PExFSeR43X6rdSJAvHjDdto2wC0Lht11uW75a9jZfBwA8OwbhzF6UG9sWDTRdB8rRXH22EEozuuJGcvro2qzsGrSbFhU7ljp9JswdeIqSuRcjPciK3BVRPlMJatcMkAKDuEaIg8gPylAfnj4u10rx4kAcXMsrDgUWWHtpttMdl3+9m5LRLnR2Nl8HH97t8XUkiOiKJ7oOMPM6NJzRlVxsqPTkdKZSsI00XMRac4qonxSD6v4QgoO4QoiDyA/KBQy440XfioWZqcaMOt7VhzKvRXFkYwiEWGdyMDUrfs+Zn7+yv6jlq4qK0VRprlo+ZC+tpXOVBKmiZ4L65rpU/5Flc9EB1t3NUjBIVzB6gHkJ4XCbLw7Draiz/n+sDD5EREllds2YGC4bYCosE5UYGow1I7zM9mPxiuG9RM6hpmiyJoXqyaNtj/rWCKW0FQSprJzcdtSzLsXZZXPVAm2ThZIwSFcweoBlOg3MCO8NzKt6WK8snySid2HW6MK5vGUVKu2AfGIn7FLTUMzKtfvZcZYjBzQSyjQWAQtFqfhYCvGFuagpCAH91w1zHG9IuO9KCtM/Xov8+YChGvs6MfrlaWYdy/KrhP1sIofVOiPCv25hlmBtERWMhUZr90ibG4+TJ0Il2CoHW8d/BSKoqBscI7ra2om+FnF2pKxemww1I4J1Vu58TFud6K2mzLP+x3x0pZFi9z5yYXMQz8X1nwnD+3nu+cM4S5U6I9ICGZvJn40zerHa6cIm5tuN5m3cta+euVDAVA9yz3hpM2TJfd5boJEv6XaURbX1DWZBv/avb5u1rcxc62aHdMvhR6dos2FN96H55T4ylJMJBZScAhXMXuYJlrosdA/MK18/EZB5ZbbzUy4WBUTC4baYywrKoCqDXtdEU7BUDv+vOcjdjVghV0NWL9OImX43XaL2LFEBEPtWLmtyfLYsteXNRYn9W147j8wgpZlxuo3F7IVvPGmKUpc44786tIjwpCCQ8QVLzKGzB4yog8gKwsTS1BNHtrPlYep3bdybV+W0aFTRZRwsvMg1s/ZSBrY1YBllQu33SJ2LBGaEifiq5e5vryxOKlvo92n+jX7xXUjUDY4x9G9mMiAZDv3Jm+8owfneGopdquSMREfSMEhkhozAcn7jvdA5VmYjFYSTVDVVU6RepjyzuvkrbwotwcUxBYeS1MQEU52LRo85YZXDVhWufDCLSJriTBT4hQA358yBL9+5YNInNaCSYWOx3KyoxP3VhRj6aZ9UinGejRXmvZfpy7gRLmQ7Sq4ZuP1ylLsViVjIn6QgkP4GivrDE9AAmB+d/zkaSzdvI/7QGVZmH5T1xSjQGhCU/RhavYg5z2sRd7K8wNZqJ41MjoG5+zxncR78Aqb3T99OK4elc/cV1a58MItImOJEFHiZo8dhBvGD8aauoNYVfcBntrWhFV1TUIxUryx7PnweOQeVAAsnjYs6h7cfbgVbx78FJcW9uEqkdohVZy7nk4Fe7xdyFb3ppVlxyrmz83xu1XJmIgvpOAQtvHa/2xlgfnk81NcAalCZX5XvWkf860LAHMuwVA7VtfFxmak4ZyFxOphKqJk8B7WIm/V2r47DrZCUYDRuiwqu0oETzjzlBuzfXhuDi/cIjKWCBklblXdB9IxUqyxLK4YFrHcAGFBuWzzflx7yQDkB7Jw9x92RbVtuHpEHpZ/u8x0zPrrybsXWYHOrPvdCxcyD7O56NdVUYDKimLcMnlIzDHi5fJ2q5IxEV9IwSFs4XVKKU8pON5+OiIg0sJenJjUbu0hw+oizXqgrnmtCau2NTHnwnuw3TS5SPjBKqpksB7Wom/V+YEsXFMS+51dJcKOy0J2H7PtnSjPk4f2wyPfugRQgbJCfsq8qBLHu347D1nHSBmvn9m98HHbF1HKDQC89HYL/vv/9uH/u6rYdMxm19P4W2U19kxE7AhvLtkZadHWEhVY8tI+QAVuuTxWybFC5l7iPdfcqmRMxBdScAhp4pFSyhME1Zv2RRWaUxQgTQ13ZTY+ZKzenoHwQ2xlbRPXl856sKUpwPyJRcJzcWqpcPKWakdR0QTC5KH9pCoPA/JuDuP2APDzF9/B6jq2wmmFjOItuja869epsq2EVoor7174856PmONc/vIBfHv84MhxZOO+jL9VVmNP1m/XSwutdmxW+44THWeYLxVLN+2LWLpEkbkfrJ5rblQyJuILKTiENPFIKWUqFoi1wKgq8PjcUvTpkRnzkGEJ295Z3aIeUt+bVBiTHmw0+TsNvnTjGE6EjYzS4cQyJ5serqGtM6uYoIzybEfxFlkb3vUbU9hHWnE1uxcuLezD3EdFdEaczPUUca2wfrteWmhZjVhHXdg7MpdgqB2KgpiaRJ2A1DNG9n4ws9Tl9OAr/KTY+BdScAhp4pFSyoxfmDYsEpypP+9ok8q9xrdnvXDIzkhD86cnLefiRvClk2O4IWxErEBOLHMrag9ErGt2xmhWTFBUebareIusjZMYKdFjlRTk4OoReXjp7Zao7Vm/LVGrnkhjT33GHeCthZbXiFVfaTg/kIXKiuKwW0qH7DNG9n5gZSQqCnDb2kaooFTwZIQUHEIaNywSIjAtMNndHJ83P5AVHcQIRN4Yecc0ChRRi4pxO9mxxrPCrF0FYcWrB7Bk0zlhZGeMZpYGUcHmteKtDzjV/raruPLuheXfLsN//98+LH/5AFQ4j+0w/lZZdKpA7XtHMXloP8vgfaf3nOg9dsvkIYAadkux3M8iyN4Pte8djfpbAQCVUsGTGVJwCCmcxGfYwcwCY/e8rJTPtLOuLjNrkIaoRcUNy0s8K8zaURCCoXZUb9oX87k2RoCdnSZyboBfLZmF14o373qKWsdEXYz/31XF+Pb4wa79trTfzM5DrRFrhJHKDXsjwtwqeJ+H2x3Ob7l8CK69ZIDtdZC5H5gWRIabTHNZTR8Vv3glwj6k4BDCyApsr370ToJuAbbS0AmgT49M19w4blle4llh1kog8NJnWQIzTQH2fHgcN6x6PXK/zBlbgAn/kstsBmo8dxrCmWrzJ4pnqwHe1XKxez2DoXasqWvCym1NUm4Op/c463jTR2Xh81Nfomr9XnQavtcL8ohVE2cVHlgXOBR9Nsgqoax1kHmuiN4PrGeCqsYqekDYZfX5qS8tC4oSiYcUHEII2Qe8n3/0TpQGUYuKW5aXeLkDNXgCgXc9e2SkM4/znfLBUfFSnSqw9s3DWPvmYW4zULeUE7eVA8De9WRVSfbazSFSHK84rydmLK83bSyqGv6fVeBQf06rZ4N+XPGORxO5H3jPBFbcnwrrgqIiii9ZfLyHFBxCCJkHvNMKpV7jRGkQVY7ctLzEu8IsK96Idz1PdJxhHuPCnNiGkhoq+M1AvVBO3IAZgApwr6dZlWSvXIyiwr+kIAfVemvZWVcMT9+xikFZczalX49+jmauPRm8jEfjPRNmjx2EAb274/Z1u5jz4xUUFVV8/fbyl2qQgkMIISOwRSuUJvLH7SQ4VEQ5ctvykkjBb3Y9een8meelMc37Gp2qecpvopVgIRT+V24ETcsgK/yN93/te0dNA5E1jMKb14Vdy8xyUynxOh6N90wwKwfwcdsXzGNlZ6QxP49n0gBBCg4hiIzAFq1Qmugft12lQVQ5irflxSvMlFvjfaEpNff/8e9m8j8mNVmPH99wWbFGqomSxg2ahnjQtOz4ZIW//v4XCUQGYpUzXgzWTZMuQn4gC/UHjrmmlMSrPIXMy4qWUWfkZIcxyilMPJMGCFJwCAlk2gawHgisCqVe/Li9fvuXOb5fXS4yWCm3+l5YdzzXGJWdxkLBuWagRhL9hivbOJMnXGOCppWw0J8/qdCTebgh/PWByHqFVcsmYr3U8Cx4888GJbuplMQ7Hk0P79nnh/5rBB9ScAgpRAU264EQDLV7/uP2+u3fzvGTwt1igZVymx/IQp/zravmKgCe//6EmC7ZGrKxXm6uq52O72bnddOCZzVXN4U/q32G2XU3O6+dcQVD7dhxqBWqqmJMYZ+obRNhFTWr0G0nKyxRSlpXRFFVC6drCtLW1oZAIIBQKIRevXolejhdipqGZmYgnxsEQ+2YWL01RoHSV0nVtrMjGEWPr8eP7hY7iKwZa30UBVDU6F5hVqUFRNbY7XWVuXfi7XKU7aeUCJeodt7sjDSc6DgTc5+IjsvYroOXcRcvRNc+GGrHzkOt6GQoZSwSdZ1SARn5zY6Ecpnly5ejqKgI3bt3R1lZGbZt22a6/auvvoqysjJ0794dF110EZ588smYbdavX4+vfOUryMzMxFe+8hVs3LjRq+ETLjJ77CDUVU7BuoXjUVc5Rcj6UX/gGIKhdstjm739a9Q0NGNi9VbMXfkGJlZvRU1Ds/DYecffcbCVOUaeu0VkLn5CdM20t9N0JRx9k64oqL5uJF6r+prw9WYdw/iG68W6itw72vjKh/R1XSjx7nPZuXo1PivyA1lo/vQEZi6vZ94nIuNiFdtTAVSt34tgqF3qWaA/puw+xvHo175qw17sPtwas23te0dx+7pG3L5ul9BzJVHXqavhuYuqpqYGd955J5YvX46JEydixYoVqKiowDvvvINBg2Ifdk1NTbj66quxcOFCPPPMM3jttdewaNEi9OvXD7NmzQIAbN++HbNnz8ZPf/pTzJw5Exs3bsT111+Puro6jBs3zuspEQ4RdXPJvqVb+bedxnewjq8AkbgT4xjdCihMpIvLaXaO3lUhipUbwotATTdiI+xeJ7P7PFmCUt2IneJlnnUCWPNaE1Ztk+sw79TKxywIqgIznqiPsiolOm6M4OO5BeeXv/wlFixYgJtuugnDhw/Hww8/jIKCAvz6179mbv/kk09i0KBBePjhhzF8+HDcdNNN+N73vof/+Z//iWzz8MMP4xvf+AaqqqpQXFyMqqoqXHnllXj44Ye9ng7hENE3Kjtv6VZv/6Jv6aLH1348vDFqQlOPrNB0YnFyAztrpn87tfsGbfaG68a6ss5nZTkyw+51srrPvZiryJjqDxzD7sNsyyQLp78tgD1XIPwSsbK2SepZ4IaVjzceFdHHcmPuhDd4asHp6OjAjh07UFlZGfX51KlTUV9fz9xn+/btmDp1atRnV111FVavXo3Tp0+jW7du2L59O374wx/GbMNTcE6dOoVTp05F/m5ra7MxG8IpMm9UVm+uvLdls7d/N97S9cf/5MQp3La2kTtGpwGFfngzdLJmXsUfeRWoaTeA1cl1srrP4x2Uyqq+7Ib1VARtrpUb9kaqLCsAbrqsKKbWjpUVyw3LlzYeVsFG/bEoM8q/eKrgHDt2DGfOnEH//v2jPu/fvz9aWlqY+7S0tDC3//LLL3Hs2DHk5+dzt+Edc8mSJXjwwQcdzIRgIWOSlxUCZg8NK8HJc4HVvnc0qjy9ItjIkdURXDQrzEnWhx/cE2YC1uz6e62cuZ1NY5YpY4WT6yQiHOOVOcSrvixy7dxSxPT1eFQVKCsMZ9utNlRLtlIg3HI5FvTJxsrvlOGm3+4wbT66YFJRZIyUGeUf4pImrijRdj5VVWM+s9re+LnMMauqqnDXXXdF/m5ra0NBQYHY4Akmsm/nskIgP5CFeyuKsXTTvqiHBmC/94sxgFFREeknY2eeog910ZgjI355M2QJWKvrHw/lzO66GnFqaXJynby+h2Qwq74scu3c7CVm7NYtqzzZfZnRMN4T140eiOcbP4o5/4pXD6B60z6oCFubbrbRIJbwDk8VnNzcXKSnp8dYVj7++OMYC4xGXl4ec/vzzjsPffv2Nd2Gd8zMzExkZmbanQZhwM7buawQqGlojig3CoDF04Zh9thBtiuj8jqIW7ULsJqnl2/XfqqZoRewIuviF+XMCjcsTU6vk18qXvOqLwNyCpsX45dZI7svM8b99ffE840fYcOicpzs6Iycf0XtASzZtC+ynwpg9baDmD+xyMYMCS/wNMg4IyMDZWVl2LJlS9TnW7ZswYQJE5j7lJeXx2z/l7/8BWPGjEG3bt1Mt+Edk3AXu4GnvCBOYyCq8QGjAli2eT+CoXbbQZd29kt06jAgn1ZvhZO0WQ2RdXEatBsv3AoQdXqd/JA2bLxmGn65dqJrZPYyIwLvnjjZ0RkVPF+tU27021FwsX/w3EV11113Yd68eRgzZgzKy8vx1FNPobm5GbfeeiuAsPvoyJEj+N3vfgcAuPXWW/H444/jrrvuwsKFC7F9+3asXr0a69atixzzBz/4ASZPnoylS5fim9/8Jv74xz/ir3/9K+rq6ryeDgH7b+eiro6CPrGdqLUHR/mQvrbelq3eslnxJH6xQvjFFaMhui5+sUyY4XYrATfmmMiyAPprlp2RFmWxSBacXlOR/ZuOnYhygWmkgd9jjYg/nis4s2fPxieffIKf/OQnCAaDGDFiBF566SUMHjwYABAMBtHcfC6lsqioCC+99BJ++MMf4oknnsCAAQPw6KOPRmrgAMCECRPw3HPP4Uc/+hHuv/9+DBkyBDU1NVQDJ06wlIXF04ZFGs+ZPQxFXB0bFpWbPmDsCk7efjzB76aLKBFCS39OwF7sEguZdYlH7IgT/OQGBPxR+drv18wKp9dUZH+eO+/eiuKkXrtUg1o1UKsG22jlxvccOR6Jl5F5KNcfOIa5K9+I+XzdwvFo/vQEt6WDm8qCSIl+p2XVEyG0jOe8aVIRnjKk2gLhtZbNGtJIpXLzfpiLnVYgqYJsRqbItk6vqdX++rYzaUpYubll8hDp8xByyMhvarZJ2Eb70d+w6nVblgEzU3D5kL5S1ha7iGT7OHmjdStd2mlK/qptTa6725L9TV+PH+bih7IAiUDmNy2zrdNrarV/Mrhguzpx6UVFpC5OgjStAo+bjp2IenB40YPIjUqxZoG7bgSxylbJ5QVZ3jTpIt8H/bICzp0GRScLiahanGh4v2lWFWU/9nbzQ3A4wYcsOIQjnAb0ydRY8eIN16m/3uqN0un6uJmSP39SIeZPKvTtG6dxLWeWDsTGxiMJjUeJJ36LB4oHvN/0jOX1UAV//zsPtSKnR2KCsgl/QwoO4Qg3HsqiNVa8ympys0R/1fq9KM7riZKCnMjcZpYOxPqdRyL7zSgdIHwOO0qd1TVxUwi4FTvBWkv9miWiTUUi6GpuD16wrir4+1cU4La1jVDRNZRgQg5ScAhb6IWVmw9lM4FuliLuNPDYjr+e5wqasbwe1WcftMFQOzY2Hona5vnGj3DPVcOEzudmSr7bmVxuxk6YVdHV6ArxKIA/4oHihVEZT0P4N6SH9/tPU8KKkHbbdBUlmBCHFBxCGrO0aqdYCXQ7bQOssCv4zd4+tQetU7eaEwuZXlC6HZwt4zqzW/nYSKrHo9jFzv2byFo7Roy1d2Y8UR9dhRhg/v6Pff4Fbl+3K+pY8VaC7a6jn9Y/lSEFh5DC6yaKIgJd1KUlMh4ngl8ba9X6vdy3TpbgTlPkioE5tZDtPtyKyvV7XX3TlVHc1hgaJbK2ZV33GaUDmP1/iHPw7l8zAeqHWjtGtN80M2BY4W9rx7rplnJhdx39uP6pCik4hBTxSGWVEehOxuOGsjZ77CAU5/WMBEVqaA9aTXBXbtgb+V5Vw80AZR5qdt0WNQ3N4XMbPnd6zURdZ8FQO1Yy6u+wlDzWdb/nqmFdJh5FFt79e/zkaSzdzK5L5fULilOajp2IuVdVld0zzo510y3lwu46+n39Uw1KEyekiFcqq2j6pZPxuNWHqKQgB9UmfZcmD+0H/VNbRXzSWyNNBxluHzdq4Ij0mmIJLCCcss66tsbrTmm4fHj3b/XZoptAbCq1W/e8V8j+nmV6gImmmYuUJrC7jn5f/1SDLDiEFH5LZXUyHjezssysTiwhH49YAV7gbhrgyjUTsbQxM18AjCoIIBhqj3I1UkyCHDz3p5lF0y/91XjY+T2LWjdFrL2iFh676+j39U81SMEhpPFbKqvd8bitrPEetIl6qPEE4MZFEyJp7E6xEi7GNdZezm9b2xgRIAB8EZNgVLL8rnSx7t/FFcMibVM09Pea315QWHjxfAmG2vHpiQ4oiDKmRq2NjPsoJvtLAb43qdByHMmw/qkE9aKiXlRdnnj0IdL3rbHqreWmYDU7rxGr8zoZVzDUjh0HW3HHc40xCpc+1RdITP8l0SKDblwbtxUn4/0rcs390HsrXuivraZgq0DM2pj1xuP1awuG2rHmtSasrG2SqsXTldbfbWTkNyk4pOA4wu9vuXbxQtizHmrxqN4r8jC1Ms27EZzJEyAsnDQBlYXV5NJIuqJg8bRh3OBdUeKVQUMCNAzr2qYpwKNzSlFWmBO1NnaanXblBqmJgpptEnEhVdMdvRL2RneO3eq9ssqVlRvJyjTvVuYHz2XGsuDo3XdeK9GiRQarN+1zlGq/+3BrVDadlxk0drPuUg1mMU4V6Ht+prXbCcACC7dTV22QmixQFhVhCz82vrNCJDvCal5uzlumeq+GbONNu+PQn9etzA9W5tWS60aiehY/G8vpfEWuOStzx0iaAm6guAg1Dc3hAnaUQRNX7GZl3XzZRYACPLWtyfS+64oNUpMJsuAQtki2NxdRq4vVvNyaNy/o0YjdIEiz8xqtIVZB0PHINmN9Fq8ijiJFBq2Cd82IpOszviNh6C12g3pX1X1gK9iYgob9BSk4hC28zAzyIghTVFD2yEg3zbRwY97GoEczJWfxtHM9q+wqV9p67v0wxIwhsXpIu/EQN15TlqBg1dCJVxFHkSKDvbO62VoDbrq+4k66vhWJiJPzU2yebFaW7H0Xj75vhD1IwSFs4dWbixdxPTsOtQo9sLRzG5UbM2GfpgCLK8QaZwLhB7++bYKKsIJzx9f+BY9u/UfM9hfmnDuuHeVKv556jALfSgg4Sd2VuaZ6weBEmbTbhV3/nfHvyUP74ZFvXQKoiAlQNYMZewR30/V5JCJOzo+xeSwFmqeE2Lnv9Mf34/y7KhSDQ9hGpoqoCF7E9dQ0NOP2tY0xn7MCWY2KQJoCbFhUHjOv2WMHYfG0YVDOjnHppn3CsSE7DrXGlqIHcEGvTGYcyG1rGyPHZsWwmCmVrDnpMcZ/5AfMqwZbfS8yBrNraoy3qX3vaNR8RYI+NdyOjdDGdtvaRtzxXCNq3zsa+c4qzocZezRrpOfKTSLi5JIlNs8stkv2d6YnWebfVSALDuEIN7M13I7r4cU+sFwDLCtPpwqc7DC20Qwfd+lmexk1vKoMvbMywo07DQqJCjlLix6rIOZ4xH+IXlOeYKirnIK6yilYU3cQq+o+wFPbmrCqrsnyrdhNC6OZu6v2vaOmb+ualWDy0H6oq5wS19RtkeBxt10oyRCbJ+K+tGuxTIb5dyVIwSF8g9txPTwB/+icUlxTMiDyd01DMyrX743ZjnduJw+xMYV9YmJuFJxzeWRnpOP2dbtMjy2qVLLWUz+3eMR/iF5TszUtzM0WDvrUowmpnYda0amqGFPYx9YceGPbeajVVFAm2lXBW/s9Hx7HDate92Rcbv2GvYxhEf392nl5o1YM/oJcVIRvcGIaZsFzU5QVnnMNyFh5rI4r8hDLD2ShetbIyA8vDUD1rJGR84wp7CN1bDP3CGs9q64uds2lKILoNTVbUydp6rXvHcXt6xpx+7pdttPqeWPrVFXuuPzgqmCt/eKKc8UKvRiXG79hL0oh6PEytdvtZxjhDLLgEL7CzT40Im4KUSuP7HHN3kDN5ijjWhGxEJhleGjn8xqRa2o1bztvxU7TzPXXkDU2TRlljcsvrgrj2sdjXE5+w6xrVrV+L3pknoeywTn4uO0LvHnwU1xa2Md2DJPXqd1+69XXlSEFh/AdRtOwE3O11cOGZ1LWW3lkjyuieJiZv0UekLKNATXF5hcvvoOV2+T65lihvz4AO7ZDxNzPm7ddgeREmLOuISuGxk2lbPfhVsfCm4Vx7ePhQhF17xh/28zKwwgH2xuZNXogHrr+Elvj81oJcfMZRtiHelFRLypfE484BpmGlDy0B1iPjHTMXF4fI0Dc7k0j0xgwGGrHb+rCwbnGX7vTsfEaGXpxrWT7K9ntEyS7H29cMvfV3X/YFdWmw4nwtsKN+92tcRh/25OH9rPsC6bnj9/3PtXeKYmOxUo1qBcVkRK41QPJCqdvc0Yhb3w2u+kC0CtSIm/iWgA1T164kaWmjUF/Dq+ulWpa9zkaO5afYKgdf97zkZTlh2etEL2vdh9ujVJugHBPsu+UD/ZEePvBhWKWNae/Zla8dbDV1wpOvJ5hBBtScAjfEs84BtmMCb2iwRPyGm65AFidx/XtBFhxQLwWAW6MzSoN3c1rZfctWEaY84oiAux1EnE7iNxXbx78lPm5l8LbzfIOdtwvZr9t7ZrtONiKO55rNL3Hxli4kuMFbw38EovVVSEFh/Atfk25jLLYnO2GbUQbt1sBjKw3wecbP8KGReU42dHJFN5WCkganLUKMEtDB9xt3eHkLVhEmJsVRWRdQzfdDpdy0tdFhbeT+A6nsSF218Hqt50fyMI1JVk40fEl15oza/RAX1hvzNbAr8+wrgIpOISruBlM53W2gx1i3DIcgWimeNiB9yZ4sqMzJuZGg6eApCnATZMuwvxJha5mqSln/XMq3E2P9fItWLtfPz3RwVRu7p8+HFePymdaxtxyO5QU5GDW6IExMTgiwtuJouVUSXOyDqK/baMF7uO2L/DWwVaMKcwxXZ94BfVarYEfn2FdCVJwCNfwIpjOD/ECeriNExHO9tAeYCUFOa6mY9vtjxPVNwvATZOLMH9ikWcpsQBcv1Yyc5cRbFZNT9MVJUa5AbxRuB66/hJcPTIPL+8/iinD+uHK4XmW+zhRMET3NVtPp+sg+tvWW+DyA1mWil88g3pF1sBvz7CuBCk4hGOCoXbsONQaFczqZjCdm/ECRmTf9HjC1mixsfOQNRuL3TfBeDxcjdfHi/gokbnLNvU0xk4pEHMtyiqbZteV1el97RvNUWP3Ir5DZF+r9XTD/eL2bzveQb2ia+DlM4zg46mC09raijvuuAN/+tOfAADXXnstHnvsMfTu3Zu5/enTp/GjH/0IL730Ej744AMEAgF8/etfR3V1NQYMOFd07YorrsCrr74ate/s2bPx3HPPeTYXgo1ZYKbfg+nsKCE8Yat/q7TzkLVbtE9/TjPlyK/XQBQrRU12zVkCXgXw2JxS9D0/09Ki4EYxRpFO72a9rpwoGLx9szPSUH/gWEzwPGs9YyyECrC4YlhC77V4B/WSC8rfeKrgzJ07Fx9++CE2b94MALj55psxb948vPDCC8ztT548iZ07d+L+++9HSUkJWltbceedd+Laa6/FW2+9FbXtwoUL8ZOf/CTyd1YW3VDxxqpbtZ+D6Zy86emFbXZGGk50nEEw1B7ZT/Yh69Rd4Mc6G17EQJgparJrblbg0S33itl1BWDZ6d2q15VT4XrTpCKs2tYUca3OKB0QqeHECp5nrefkof0we+yFWPvmYXSqwNJN+9A7qxsmD+3n2vWXuZcSEdRLLij/4pmC8+6772Lz5s14/fXXMW7cOADAypUrUV5ejv3792PYsGEx+wQCAWzZsiXqs8ceewyXXnopmpubMWjQuYd2dnY28vKs/dSEd5hl6fj9Tcbpm15+IIv7di37kHXiLvBjnY1EKFyya+7Gm7eVZczsuqqI7WOlx6rXlZP4DuP1uXnSRZg+Ki+qQCUveF6/niwLVKcKVK7fC+XstUhTgHunFWPkhQFbyo6de8mouMXjOZQKVtJUxDMFZ/v27QgEAhHlBgDGjx+PQCCA+vp6poLDIhQKQVGUGLfWs88+i2eeeQb9+/dHRUUFfvzjH6Nnz55uToGwgCVU0pRwHyfRN+FE4fRNz0qxkBGeVmMxO5ff6myIKFxeWXdkFRav37yZvw8An5w4hYKcLMtO72a9rvTohavV2rKuz+q6JowqCAgFz+vPw7NAqTinIHWqwJJN+8LHklR2ZZV3luLmNFOQSG48U3BaWlpwwQUXxHx+wQUXoKWlRegYX3zxBSorKzF37tyoksw33HADioqKkJeXh7fffhtVVVXYvXt3jPVH49SpUzh16lTk77a2NsnZECx4QoXXpNJNnApJp2/wVoqFqPDU5nHvtGIs27yfORazc/mtzobVunhp3eGteaLik4z3mJalddvaRmahxsUVwzBqYG/hXldGRNaWd32gxvaoAoD/mDIEk/6lX8w9bFVjiYWsdVFGeecpbvMnFcoNkkgppBWcBx54AA8++KDpNg0NDQAARVFivlNVlfm5kdOnT2POnDno7OzE8uXLo75buHBh5P9HjBiBiy++GGPGjMHOnTsxevTomGMtWbLEcsyEPZy+BdtRVNwSkk7GLqJYWAlP4zzurSiOEXBW5xJR1OLZ6M9srPFwpxnXPBHuMv1686rydqrWhRoB8XtUdG3NYo/urSjGkpf2RR33yVc+wLfHD445L6/Gkr4fGQsZ66KM8u43SybhD9Jkd7jtttvw7rvvmv4bMWIE8vLy8M9//jNm/6NHj6J///6m5zh9+jSuv/56NDU1YcuWLZYNtUaPHo1u3brh/fffZ35fVVWFUCgU+Xf48GHxCROW5AeyUD6kry3/+sTqrZi78g1MrN6KmoZmy314D/JgqN3O0G2PXVMs0s8q67IWINY8lm3azxRiVueaPXYQ6iqnYN3C8airnBIlwGXWOBhqR/2BYwiG2qP+XwazsZoJIS9w+14RgbXe+YEs9Dk/gzl3rVCj2X0jco+Krq3Z9Rk5MBBzXN71MR4nTQFuvuwi1Fd9DdWzzn1uRMa6KPMb65GRDuMZ/ZzkQMQHaQtObm4ucnNzLbcrLy9HKBTCm2++iUsvvRQA8MYbbyAUCmHChAnc/TTl5v3338fLL7+Mvn3ZVVr1/P3vf8fp06eRn5/P/D4zMxOZmZmWxyHih923eT+9qTmxAMnOw+pcLGuRzBq72RWcN9Z4u9PcuFdkrF9m6+313GWOz7o+wVA7Pj3RwSx2yBsj7zpPHtoPD88pQZqi4MPj7Vi2ie16FUHkN6bdu8Zx+znJgYgPnsXgDB8+HNOmTcPChQuxYsUKAOE08WuuuSYqwLi4uBhLlizBzJkz8eWXX+Lf/u3fsHPnTvz5z3/GmTNnIvE6ffr0QUZGBg4cOIBnn30WV199NXJzc/HOO+/g7rvvRmlpKSZOnOjVdAiXsSt8/BZzYjeGw25lYqtYHr0gFl1jL7qCs8aqvZHrXUZeCiGn94qse8tsvcuH9PW0XopsTJn++vAqOouMUcQlWFc5xVEgt9V9bwx2TlOADYvKfdGnikgsntbBefbZZ3HHHXdg6tSpAMKF/h5//PGobfbv349QKAQA+PDDDyNFAS+55JKo7V5++WVcccUVyMjIwN/+9jc88sgj+Pzzz1FQUIDp06fjxz/+MdLT072cDuEidoWPG+m9fsDNefAEsegar6lrMg0YFVE8ZSwdWoYNKxXZTXhrDAD1B46ZjtWOhdFqvb3O2rJzfJZym6aECx7KZkLy1qyucgq3X5pTWEplpwqc7Oj05HxEcuGpgtOnTx8888wzptuouqdcYWFh1N8sCgoKYqoYE8mHEwHPM7HHK5DWLdwQeE7T1YOhdqzc1mR6DivF06xGj/6aaGPVfuEqvK/ZY1zj2veOYmL1VkurjB0Lo8h6e5m1Zef4PAWh7/mZ0oHqiXAf+82iS/gL6kVFJAwnAp5nYrfbFdlL5cjLNGWn6epNx04wM15YbgrWPHYfbkXlhr1RdU/u2/A2jp88HemtpF2Tgj7ZCYmf0tZYxipTlNtDKh5FI9mq2orUYFpT14SV25osY7K8UDasfpupYtElvIEUHCKhOBXwTlOPvU4j9ur42oO/R0a6o3R1XjG6jd+fYNk8FEBUg1WNM6qK6k37ohqvVq3fi43fn5DQt20ZC0Pte0ej/lYgHi/ktZXGTcwUBF6lYt7vy21lQ/S3I1tzKpmsvIQzSMEhkhonZnGv67J4dXzjg99YME5EqOgf9naah1adtdrwHMrGzzsBvLinJaFv26IWBqMrDQj3ZirO62kZu5OM8Fy+dproumXBkv3tWCmVfuzXRngPKThEUuPELO51zIAXx2c9+EUKxumRzXThxWnIsqruA7xW+TXHWTV2yQ9kYWbpQKzfeSTy2YzSAcLznfFEve3UeR5+sSoYFQSrPnN2s/1EcfO3w3OjJrJfGxEfpAv9EYSf0Mzi+mJgi6cNQ9OxE5YF3TTlSI8XtUncPD7vwS9SMA7gvxkD4O7PmodZLfJ0RcHccQUxn3eqiAgoO8UVzWAVJjR+Fgy1Y2Pjkaj9nm/8KOY+Yc0XQJTLrXLDXscFA+0UuowXvDUA5FP77RSNdPLb0Z+vpqE5rJhylCUitSELDpH06M3ie44cx9JN+4RM0W7HDBjfxr0IgHQayOlWdtCCSYV4ipF9lYawAJw8tB+ee/Ow5/E2wVA7flPXhNVnU9318UFGK5VokLNxvqyWBKoK7DzUiumjxFKxjVYaP3aB15MfyGK2bkhTwoX8zAiG2vHWwU+hKAqOtLbHBJuLWL7s1kwyWid5blSv7kU/WOOIc5CCQ6QE2gPlhlWvSwkNt2IGeD5+t7NqnCpNbmUHAcAqQ/2cNAXYuGhCJH7H63ibmobmmCBnVnyQdh9sWFRuq9rv+x+34b/++E7MNiJ1fHj3hZ8qcvNgtW7QW+FYsK6Jfl9ZJU6mZhJLaWThRYFJivHxJ6TgECkDT2i8uCeI6aPyTS0UXmZyuZ1VY6drtoab2UFWwcmzxw5CcV5PNBxsxdjCHFcry7ICgTV4gbEnOzptVfstzM3Gj//4TnTQMYCyQvP5JLJ1gxvIjjEYaucqNxqyCQAyNZNEOpynIVoJdwO/W+O6MqTgECkDr8Pxz158F7946V3P3qoS8TZuVDZE3iB52UFWLgceVtYpL99qzYQZyzWhCebyIX2li0TmB7JQPWskqtbvRSfCQnLJrJGW1zaRrRusEFGGZa2FvJpKerxMAGD9/hUFUNRwFh9LCXeDZLDGdVVIwSFSBuMDWY/+rQqAq77yRL+Ni75B8rKDnDyIedYpr99qecqsPgaHJ5j1Y3a71orVGHmtG7Iz0nCi4wyCoXbPhaKM4ikzb5b7U4+MEme3VxtLIXO78KJROUz075/gQwoOkVJoD+QX9wTxsxffjfrujKpiTd1BrKr7QOjhLho0mIhqqvqxib5BxvNB7PVbbUwgMICbJhdh/sSiyPGtBJvbtVasxshr3VD73tG4xW/YUTxF561ZuvRuKkUBKiuKMWpgbykFw+5viqeQufVb5CmHVE3Zn5CCQ6Qc+YEsTB+Vj1+89G5MhV5NuQHMH+6y7pV4lug3ju3eimIhxSWeilg8lCmrNbcSzF4pYXrl02qM8Y7f8Frx1Oa742ArFAUYPViuYSfrWLK/Kbdj3jTMrlWytejoKpCCQ6QkoqnNrIe7XaHj1YNVD2tsyzbtx73TirFs835LxSVeD2JZZcpuiq2TNbcTRGs1Rp5iLBMYy7sn3XCrxkPxzA9k4ZoSd+6rePymRLG6Vn4aKxGGFBwiZRFNbc7OiK536eegQd7YRl3YW7hCsPFBrBee2jnciE8SVaYSlWKrKWHGWisAYloyyARxyyjGIgqH2bllFR+vrHhe1IDxW10Zs2vlt7ESYUjBIVIas9RmIPywmrm8Pkpo+Dlo0Gxsdt4g9cJTHyCqAKie5VzRsBpTIlJsjQqdvtbKm02fxigTk4f2sx3EbaeIol7hMFsfu7E7blvxvFBQ/VhXhnet/rTro0hzWb+MlQhDCg7RpdBqs2h9hQB23Rq/Bg26OTaj8NTLZhXhTuFe1/KwW7vILkaFDkBUrRV9nyrtvnh4TomnQdxmCgdvfXYeanWkGLrlTnGioPKsHn6uK2O8Vn/a/RGWbDpX7dlPYyVIwSGSGH1J+DKJYMYTHWdiUlk1oZHTQyw4NJG4NTarwmgqgB0HWyPxFF6Y4b2qXSTSHsGqZguAs60aFM+DuHkKB09p6lRVX7hR7bpzzSw0fnARm93r2rUKhtpRvWlfzL5+cWcTpOAQSYqxJLyMS4VXEOy2tY0xZma/PqTceAPnKRd6zvYw9cxlYFQK9Nh9G5Zpj2BFuqJg9OAcYcUlXq05xhT2se1GdVNRtWO1srLQJNpFLHqvNx07wWwhkQb4wp1NUDdxIglhlYRXEe5BJNKxWBMaWgfytLPBJ3qXleixkhnjOhjR0nx5AsnJ+mgdn3cfbkVBn2xsWFSOH00fHrOdbNdn1lir1oevJa8ruvZZuqJg1uiBUZ3pNUVm8tB+eHhOCZ6YW4q6yimmyl1+wN1u6bPHDkJd5RSsWzg+cm7jtRO1FrndwdzOOMwsNHaP6RZm97qxKzqv4/q9FcW+fTHqapAFh0gYdt8keSXhZary6t+0j33+BW5ftyvmWGtea8J9V39FeFzJiFUndgD4856PXHUZ6N+QNWTq+ZjBrNYMYE3dQdw3fbhQpdt7rhoW9bcfAl5ZFjtZa5FXsS2y4xCx0CTKRcxTvta81oRV25pi7gFj1/l7K4pxy+QhcRkrYQ0pOERCcCI0eCXh0xQ507Del8463qrapqjKuKkAS6nU1qF8SF9cWzIgIlRq3zuKidVbmW4do0ASVVaNQlZDtp4Pj71HQszPV9V9gPmTCiOCc+ehVnSqKsYU9olRHvR/u6UUeJVGLOOq9DK2RWYcorFKiagrw1K+0hRgZW0TMynBz7F6BCk4RAJguhE27EVxXk+UFOQINz80loRfcp11A0QWxg7bGp1w1qfJb4golXqlj6WIALEuAxll1SwORraej5FgqB1LGUGfQLR1r/a9o5F7xyp2y8qdIqK0+MECBPir/IFfFQOW8vW9SYVYaVIgNBGKGCEGKThE3OE1fZzxRD2uGz0QGxuPWAoDt0rCszpsaySy9o3bb/yylgieInL/9OG4Wpe+LXtcs8BmVj0fmXUwU570Bdn0irFVOjxPKdjz4XHcsOp1y/uUFxPUI/M8qcw/N3C7/IHTe9SvigGrQOhqQ4FQv9TFIswhBYeIOzwhx6tDYtb80GlJeJ5QTAMcBzbaFQBevPHLuid4gv1qQ20a2ePysqZYwlZ2HbgdxnHuWr6w+0iMMmtMhzcbb7qiYHHFsEisEmB+n/Jigm5b25gQa45bHcz9YpXyCqPy5de6WIQ5pOAQcUcTGjwXiB6esJRRHsy25fncNy6agJKCHKl56bErALwKBJV1T4i+7dtxexiF7MmOzhg3hd2u18agz5smXYT5kwoj+xxvP83c93h7h9B4C3OzpZQ6M4tVvKs262OunHQw93MhPjOcWJz86lIjzCEFh0gIrIrCLFjCUkZ5sNqWJ8idKDdOBIBXgaB23BMiD3W7bg8r94TddbAac052BnM/3ue88bKU4mOffxFjDeFZrGTmJIsmyPceCcVkxc0eO8ixguKHQnyyGCtYV1YU45bL5bKd/OpSI/iQgkMkjJKCHFTPihaOM0oH4PnGj7jCUvThrFU5FtnW7bczJwKgR0Y683NjQ1A7iMzT+JYr8lAXXT+ZN2i7BeS045cP6cvcpmxwTkzGnBbDJYpRaVGUcB+r29ftYirR+nixO55rtJyTE0sDKwUfOHfvF+f1xJsHPxW+P1ljcTNYOR5NKlkVrJds2gcooJTuFIcUHCKhsISjsQ6JHhHlgfeQ12+rHUtGkIviRACc6DjD/PxkR6crYzObp5O4Cqv1kz22rGVI9A1dy8AzjsWuy2LnodZIBWyAr0Rr8WInOr40nZOTa2CW+QaE732exdTKUqoAWHhZEeZPKnItWDlecTy8OLulm/bh2pIBZJVJYUjBIRKOUTiaCUsr5cHqIS+TAeOEBZOKIpkXMgIgUam8XsZV2D22jGVI5g3dqcVOb3XI6ZHB7GvGs9aZndsL15ERnnJjZSlVATy1rQmr6poivxenaxivOB5e3SyZwqBEckIKDpFUWL09WqUKy2TA2MH41nvz5CKhYoF6oZmIjA2v4iqCoXZHlZBFLGt23tDtWuyMVgc71Zd553Z6DcwCmnmf/+DKf8G/XHA+xhT2sRwLEPt7sXtvxDOOJz+QhcqK4qiu30DYNelXtxrhDqTgEEmH2dsjMysKwGNzSzF6cI6nD1bWW+/qbQcxf2KR6X4sU73dYnd28cJyZOYqdNMqFa83dJbVYdmm/bjqq3nY9HZLZLsZpfbcHk6vATOlfdowjLqwN7Iz0jBzeX3UsRUFePRv/4hpMMsbi4Ybv5d4WyqvvWQAqjfti75HLKxdLFI9PT7VoGabRFKSH2A3NNQe8vpGfUtmjcT0UWGhw2qQ59aD1arqLQueqR6ArYaNxoaAotuw1s1pETjRSsii4+ahvaGzzuOmwORd38065QYAnm/8yPY8llw3MvJQtlOLydiY85bLh6B8SF+UFORYNpjVN1A1jkUPb11lrqHb9xsPbUw7DrUy6x85beTqtOks4S2eWnBaW1txxx134E9/+hMA4Nprr8Vjjz2G3r17c/f57ne/i9/+9rdRn40bNw6vv/565O9Tp07hnnvuwbp169De3o4rr7wSy5cvx4UXXujJPIjkwszC41aAJAs7b6VOLUp6c7lIbROzN1A3s8lkKiGvqWvCym1NTEuCKLdcPgRQEHE/eiEweTWTXLcIauYodpN3S3iuI6sGs8Zxa9uvqTuIVXUfcNc1GGrHb+qaIjFnotfQ69oyRnex0cpnjN2zcjslY3p8V8dTBWfu3Ln48MMPsXnzZgDAzTffjHnz5uGFF14w3W/atGlYs2ZN5O+MjOgaFXfeeSdeeOEFPPfcc+jbty/uvvtuXHPNNdixYwfS09lptkTXwiw+wKsHqx3lyYmp3qisqIw3cn1s0e7DrVFtCljbuJVNJlIJmeXCchITdcvkIVHNQr2I5bCqagw4S5nmBd4CYn2vROaQHwj3GmO5co3jzg9k4b7pwzF/UiFzXWsamqPuKeO43YixsgPLXazgnEKq/22Kup381MuLEMMzBefdd9/F5s2b8frrr2PcuHEAgJUrV6K8vBz79+/HsGHDuPtmZmYiLy+P+V0oFMLq1avx+9//Hl//+tcBAM888wwKCgrw17/+FVdddZX7kyESgpfBfF49WGWVJ7sWJZYwNKJ/u6xpaEYlo+eWl4GdZvMyc2E5GZNX11WDdX17Z3VzxSLIsxCsea0Jq7bJWUes0K6PsS9X7XtHmcdmratZH7dEWzZYa6kCeGxOKfqenxm5djLZXF5afwlv8EzB2b59OwKBQES5AYDx48cjEAigvr7eVMF55ZVXcMEFF6B37964/PLL8fOf/xwXXHABAGDHjh04ffo0pk6dGtl+wIABGDFiBOrr65kKzqlTp3Dq1KnI321tbW5MkfCQZA7mkxWyWlXnhoOtGFuYI1RFWSQlWN9gsmrDXjAK6bryBspTRM3mJdIY068Yr69bFkGeC2xlbZNlnR07TB7aL1KkEAgrAG6lpSf6GvKsLWWF0Q1OZd1O1LIhufAsyLilpSWilOi54IIL0NLSwtgjTEVFBZ599lls3boVDz30EBoaGvC1r30toqC0tLQgIyMDOTnRQqB///7c4y5ZsgSBQCDyr6CgwMHMCK/pasF8NQ3NmLm8Hj978V3MXF6PmoZmy31YwdKKcu4HrX+79LKhaE1DMyZWb8XclW9gYvXWqLGbzYs1fv2YANgOOk4EvKB32WMYA28XTCriWt2cYicoXg/vGioAFk8bllDhLxrEbCfpwI1rTcQHaQvOAw88gAcffNB0m4aGBgCAosTe/aqqMj/XmD17duT/R4wYgTFjxmDw4MF48cUXcd1113H3MztuVVUV7rrrrsjfbW1tpOT4mFQM5jNaObS/e2SkxyhzVRv2ojivZ5TFg9VCgWUuZ71detVQ1CpmxMz0bxy/vjFm7XtHMbF6a1Ja75xitBAAiATvarhlHXE7LV17+qoAlm7eh97Z3RJ63bzspUYkB9IKzm233YY5c+aYblNYWIg9e/bgn//8Z8x3R48eRf/+/YXPl5+fj8GDB+P9998HAOTl5aGjowOtra1RVpyPP/4YEyZMYB4jMzMTmZmZwuckEkuqBfMZ3W0zSwdiY+ORcHaHzkWg0akCM56oR/WssGDnuet4D3DR2AFR5YbngjJTRFWolkoqa/zJ2qnaTYwuMK8EsBvCnddnyy/XTcRdbMdFTCQH0gpObm4ucnNzLbcrLy9HKBTCm2++iUsvvRQA8MYbbyAUCnEVERaffPIJDh8+jPz8fABAWVkZunXrhi1btuD6668HAASDQbz99ttYtmyZ7HQIH5JKb1Usgb1+55HI96y4GOBcPERxXk9LS4jIutiNHTCLhbJSREWUVOP4U9F65xSra+ckGN+NmJL8QBb6nJ+81y2Z4/0IczwLMh4+fDimTZuGhQsXYsWKFQDCaeLXXHNNVIBxcXExlixZgpkzZ+Lzzz/HAw88gFmzZiE/Px8HDx7Efffdh9zcXMycORMAEAgEsGDBAtx9993o27cv+vTpg3vuuQcjR46MZFURyY/+wZudkYYTHWcQDLX7/mFpRCQYGIit0QGEBUTDwVbXBIds8LOVNcVKEbWjpHrZTT2Z4V07N4SzG5lnTq2uiWp/QBbD1MbTOjjPPvss7rjjjkjG07XXXovHH388apv9+/cjFAoBANLT07F371787ne/w/Hjx5Gfn48pU6agpqYGPXv2jOzzq1/9Cueddx6uv/76SKG/p59+mmrgpBj5gSyh4nVm7D7cijcPfopLC/skxPRsVvJeI11R8NR3RuOm3+6IKUQ2tjAnYe46EWuKmQXAjnXA627qqYSfhLMTq2siLShkMUxtPFVw+vTpg2eeecZ0G1Vno8/KysL//d//WR63e/fueOyxx/DYY485HiPhX5w+wO/+w64od9Cs0QPx0PWXeDRaNqwH/4zSAXi+8aMoQXDl8DxUz2LHySTKXSf6Vm5mAZC1DvB6iWVnpKH+wDFP3vCTtXmi34SzHYU20UpaqsX7EdFQs03Ct1g9wM0E0+7DrVHKDRCOfflO+eC4W3JYD/57rhoWIwh4AiJRtTfiEQvFyw6LqriMcNC1k1YOPOJtPXBTmfKjcJZVaHm/8R0HW3FNiff3OSubb3FFYlPcCfcgBYfwLWYPcJZgmjy0X0R4vHnwU+Yx3zrYmhBXlfHBzxMEsp/bRVTQeqlc8ZQLYwE6wLwFhV3ibT1wW5nyi3CWVdr02/NcuHc814gTHV86Wh+Ze/z4ydOoPttyY+mmfeidldgUd8IdSMEhfAvPggDE1lip1FXqVQAsmjKEecwxhfFTbvzq+pAVtG4rV4C5cmEVmO2WG2bHIfcCuK3wSplKtHCWvZdY2y+5biSq1u+FPsrK6frIjCsYasfSzfs8UaKJxNK1UxMI3zN77CDUVU7BuoXjUVc5BbPHDmL3mTG87S9/+QCuHhHdz2zW6IFxs96YVfhNJG5XiQ6G2m1VHDZzP/Iq5Gq44YapaWjG7WsbPTk2C6dVg3nwhHM8KkCz7qWq9Xvx5z0fMc9vpuQ9Orc0Znu76yN7j3t1bYjEQxYcwvcYLQgimUkqgKtH5uOWyy/CWwdbUZibjayM87ip5m5aWxIdOGmGW4GpwVA71tQ1YeW2JluxMWbux5gKuWdz6FXwS+5bjdVYRZrVJFKB89YVPLyKl0lkoDHr3J0AblvbyLwfzMZaNti9bEHZNfFjLBPhDqTgEElHfiAL904rxpJN+0y3UxSgpCAH+1o+w8Lf7eCaq3nmbCulx06F30QrOG48zPXrpSGrxMXEjwBYMKkw8j2rZYGdOCDWtS3oky1Um8gJoq01kjnQ2OxFg3U/yCi1sutjFddjtiapVFiUiEZR9XnaXYS2tjYEAgGEQiH06tUr0cMhbFB/4BjmrnyD+72iAPWVXwOASF8jjXRFQV3llMjbvPH7NAD/MWUIfv3KASmlSIsf6ZGRjpnL67nnTDQ1Dc0xD3NRywtrvfSsWzge5UP6Co8lbAk6iFV1H7ieycQaa7qiYMOi8pjro//e6XUyi/8IhtpdD9h2cj3dPDcL4/1gNVY768NabwDSa+LFtSHcR0Z+kwWHSErM3h61h1x+IAv1B44xrSk7D7Vi+qgsvHXwU6aZ/YmXD5z72/A2ynJBVW7YG3GjaP2mjLVu/PLQFMmMkrFOadi1HGjKDeCuO49nSTvZ0Yl7pxWjetM+bqduu+cWqf7s9n2QqDIC+nMbe1EB7PvBaqxuVduuq5yCusopUmvixbUhEgspOERSwjIrL64YhlEDe1t20gbCcQJb932MjY1HGEePRS/4rIKcO1Xg+caPsGFROU52dDpqNeFVJpbZw1y2/5SGnRRlnhKiKaBO4Lkq9hw5HhWYq8epeydR7slECuf8QBauKcnCiY4vhdw8bo7VbL3Lh/QlhaWLQwoOkbSIvLlqipAxZkQFYgoBmqEXfCJBzpqloPnTE7ZrnySihL2IBWLBpCKs3NYUs++ogb2lz2emgH5+ylkdFKYSPG0Ylp5NqTbihqWtKwesOrEk2VXku/J6E9ZQmjiR1OQHsizf1CYP7Yfvc+riiGAUfJrgTFfCucxpSjgDx7hPdkaa7ZRst9O5RRFJmf3epCLmfO0IFW0tjWnhKqznK5KibiwzMPLCAFO5uX/68EgZAicY7w2/uSe9RuT3aIRVUkG0/EBXX2/CHLLgECkNK+OHh4JwcHKnCq7LS8P4tlr73tEY8/yJjjO23RWJcnWIvBHnB7KYfbN447J6O589dhCyM9Jx+7pdUZ+bzVfGumV0ibDmd/WofNfKByQyJibZYMazrd8b+R2KWC5pvQkepOAQKYvx4WnFoilDUJzXE2mKgtGDcywflHrByXrIrqg9ELOPqKUjUaZ3XmxT07ETke8BcaEiqoiMKewjPF9egbkemeehzOK6yaQEO3ERUsCqGMx4NpyLaRMNOqf1JliQgkOkLLyMn3njBuH3b8RWFl7+8gFHDR31D9lgqB1LGXV6Fk8TC8RNZG0OvfKy58PjkZgV47pYCRWZgocy85UtMGc2P7MsMt7YtTFQZ3PniMazaS7SrrQ2hHNIwSFSCpGCX/825kI8+2Yz880RcCdVmadcjbqwt/AxEml61851w6rXbadwy7rZROcrW2CONz+z73ljX/NaE1Zta/Ik8DsRQeWJxqjYstAy37R7sausDeEcCjImUgZjsGLte0eZAYglBTkxQcJGnPaiYfVTsuNishO0KYNZMKfTHj121kBkvvmBLNxbUcz93o0+QqyxpynAytomTwK/ExVU7ge0QPDHv1XK/C3eesVFUZlvXWltCGeQgkOkBGYuBWOzTiA6u2bjogmuKCN6kiG7w6ohqFMlzcs1GDkwwP3OjVgl1tgXTCriFgZ0ip8bPtptqCqzb34gC33Oz2Ba5XKyYz/3y9oQ/oZcVERKYKfgl95NwaqXYgyslcWOiyleMRgi8TFuxAF55WbjuanSFPcaZrL6Ya2ua4pxeboR+O3Xei5O3Gay+/bISMfZvqoR0hUFYwvda8RJdC1IwSFSAqcCIiqw9gg/sFYWmeyOeMZgiMbHuKGgeJHhYlS+0gDcNLkI8ycWeVrx2avA70QGlfOQCRJ3uq927xuVG71L2a2yBETXgRQcIiVwQ0C4EVhrFyfCxA4yCqEXCopdIaTfT1T5clPgeRn4bffYovOTXQcntZhk9mWVc0hTgA2LylFSkAPA/bIERNeAFBwiZXBD+CSqwF68z5tIi4FdIcTr4K4yu0o5O5cZXtZckT22WVd7vSJjZx2cWEVl9mWm/avAyY7OqM/cLEtAdA1IwSFSClEBob3N9shIx4mOMxFhkKhYiEScN55p6Pr1tiOEZCreOj1XsiDS1V5TeOysgxMlWGZft+79RL2cEP6FFByiy8Fq36AXBgsmFUWCSeNl2UiURSUeFWD1620MIgXEhJBoxdvjJ09j6eZ9js6VLIh0tb9vw9t4eE6JbcFvpgSLtOAQUaDduvf9GqhNJA5ScIguBa99g/HtVwFwswdBq2Y4taj4MbjSuN4sZ5KIECrK7cFUWPScUVVUb9oX2cbuueKNm520jZxRVaQpii3Brx9X+ZC+Ud+JurxEFWi3gtn9FqhNJBZScIguBa/CMBD99qsCWL3tIOZPLIrLuDTsWlT8GlzJW29N4LophHjC3otzuYXTfldRmWRK+B42ZiKNHiyXhWQ1Lq9iXdywJlLjTUIPKThEl0LkrVcjWdwZfg6u5LkNNiwqx8mOTmEh1HTsBNMik4ZwHyqtKai+4q3dc8ULN66bVVd7rVGqVvBSRPBbjcvvsS7xcLsSyQEpOESXgtf7hqX0+NGdwcJvAsfoclly3UhUrd+LToQVEq22iQwsF5WiABsXTYhSXnpndYuxVMieS2ZuTnDruukFumijVCfjolgXIlkgBYfocuiFQN37R7H8lQORoFQgLET96M7g4SeBw3JtADgX8cvoNWQbFbigV3ehwoRuKCZO3EnG8wdD7fj0RAezcq/d66Y/R2FutmU9J96aWN1PFOtCJAuk4BBdknNF/Q5EBaWmKcBjc0pRVpiTNA9svwgclmujav1eQCcsNUFbnNczKj3fCpaLSgWY1g6ji8KN+CQn7iTj+WeWDsTGxiMRpVpTcpxcN+M5FkwqMrXCmK2JyP1EsS5EMkAKDtFl4RUY63t+ZtI9sP0gcJjrCcSkM51RVcx4oj6qVouZwhEMteOTz0/ZzgRyIz7JrjuJdf71O49EvndDqWadY3VdE9c6JLImIvcTxboQfoe6iRNdFqfdsv1GfiCL21g0HrDWMw2I+Qw4J3g14crrNq11PL993S6oajjuBhC3drjVpdvuvWKWtachq1QbO3TzFPWFk4uYndxF1yTR9xNBOIUsOESXxS+uHa+Id10c3noCiEplttujSAWQpgKPzy3F6MFi1g634pPs3isiWXsy4+G1ZmDNcf7EcB0noxXGTzFbBOElnlpwWltbMW/ePAQCAQQCAcybNw/Hjx833UdRFOa///7v/45sc8UVV8R8P2fOHC+nQqQos8cOQl3lFKxbOB51lVN8UTvGDTTLx9yVb2Bi9VbUNDRHfW+0ArgFaz31n21cNIFp0dlz5HjMuHgurz49xK0dmmLCsmS4MTc75581eqCt8fBcSwC4c2RZYdxcE4LwM4qqqhYGVPtUVFTgww8/xFNPPQUAuPnmm1FYWIgXXniBu09LS0vU35s2bcKCBQvwj3/8AxdddBGAsIIzdOhQ/OQnP4lsl5WVhUAgIDSutrY2BAIBhEIh9OrVS3ZaBOFrgqF2TKzeGvOGXlc5xTLANB6sePUAlmzaF/WZsY5NmgLcW1HMrGujzUOGYKg9ofFJxvPbGU/9gWOYu/KNmM/XLRyP8iF9pY+Z6DUhCDvIyG/PXFTvvvsuNm/ejNdffx3jxo0DAKxcuRLl5eXYv38/hg0bxtwvLy8v6u8//vGPmDJlSkS50cjOzo7ZliAI67iTRBcFHHlh7ItIpM2CblzLNu3HvdOKsWzzfscuRJGAWC9desbz2wnQFUnftls/hyBSEc9cVNu3b0cgEIgoNwAwfvx4BAIB1NfXCx3jn//8J1588UUsWLAg5rtnn30Wubm5+OpXv4p77rkHn332Gfc4p06dQltbW9Q/gkg0XrmJzAJi3Qq6dXt8aYhulaGNa9SFvePiQrRy6fkBci0RhByeWXBaWlpwwQUXxHx+wQUXxLihePz2t79Fz549cd1110V9fsMNN6CoqAh5eXl4++23UVVVhd27d2PLli3M4yxZsgQPPvig/CQIwiO8dBNZBcQmOsCUNb7F04ZFuoAbx+W1pcFOKnmiGptapW/7seEqQSQKaQXngQcesFQWGhoaAIQDho2oqsr8nMVvfvMb3HDDDejevXvU5wsXLoz8/4gRI3DxxRdjzJgx2LlzJ0aPHh1znKqqKtx1112Rv9va2lBQUCA0BoJwm3j0juIJQr9kjrHG1zs7ts2Cm+PiCX/ZGjcyyqkXCgdP4Ut0bBVB+A1pBee2226zzFgqLCzEnj178M9//jPmu6NHj6J///6W59m2bRv279+Pmpoay21Hjx6Nbt264f3332cqOJmZmcjMzLQ8DkHEg3j1juIJQj8UBQRix+fluMyEv0zatIxyGk+Fw88NVwkiUUgrOLm5ucjNzbXcrry8HKFQCG+++SYuvfRSAMAbb7yBUCiECRMmWO6/evVqlJWVoaSkxHLbv//97zh9+jTy8/OtJ0AQHiDzpu6HOiR+DTD1YlxWwl/GqvXWwU+FlFNm24oNe3HqyzNQoKB3djeMKezjaK76e85vDVcJwg94FoMzfPhwTJs2DQsXLsSKFSsAhNPEr7nmmqgMquLiYixZsgQzZ86MfNbW1ob//d//xUMPPRRz3AMHDuDZZ5/F1VdfjdzcXLzzzju4++67UVpaiokTJ3o1HYLgIvum7hc3UVdBRPiLWI+062yEpZzyqgv/1x/fifytAKieZc+qY7zn7p1WnHClmSD8hqeVjJ999lnccccdmDp1KgDg2muvxeOPPx61zf79+xEKhaI+e+6556CqKr71rW/FHDMjIwN/+9vf8Mgjj+Dzzz9HQUEBpk+fjh//+MdIT0/3bjIEwcCua2Dy0H54eE4J0hTFsiqv3wJH/TYeK0QtZkbrkX6eQHR6vUYawFRORSoYqwg3I5V1I7HuuWWb9+PeimIs2+Q8pZ4gUgVPFZw+ffrgmWeeMd2GVWfw5ptvxs0338zcvqCgAK+++qor4yMIp9hxDchYfPwWOCo7Hk1J6JGRLtU93E3sWMyM87yJ0Z0bAB6bW4rpowZEfabNWa9w8OgEuyO6Gbx7btTAcEp9omOrCMIvUC8qgnCAbDyNjMXHb4GjsuPRKwkaicg6AsQDmIOhduw41IrK9XujGoKu2tbEvM6jB+dE7c9yHV2Yk4Xvr21kni8NiLpXROZvds/ZiWESXfNks9wRBCk4BOEAUeuAJhw+PdEhbPHxW+CozHiMypBGvLOOjEJZ1GpjpBPAzZMuwuq6Ju51ZimASzftw8bvT8DSWSOjlCYgHIOzZNbIyDFE5+9mDJfoOf1mSSQIEUjBIQiHsKwDesFa+97RiHBQEP6nF3Q8i48fsq3sjoelDGmIZh05tVbphbICoLKiGLdcPoS5LU8h00hXFMyfVIj5kwq5ViBec9AZy+tRfd1I1Fd9DTsOtuJ4ewd6Z2WgrPBc7JXs/N1IqRc9p98siQQhCik4BOECeuuA8W1XVc8pNCrCwlZTFMzevv2WbSUzHrMgW9GsIyfWqmCoPcpiogLhBp8KcMvkWCXHTCEzzlPGdQSEr/99G95GXeUUXFMygLmvnfnbTanXlO9PPj8ldE6/WRIJQhRScAjCRVhvu0ZUAI/NKUXf8zMt3779UpRPP57ivJ5oONiKsYU5KCnIYW5nVIY0eEqRmXXITuzHjkOtYOkrSzftw7UlA4TOn6YAj84pjbK0mKHNuWr9XnQavrNSCHjzz85IQ/2BY67FvRiVbxFrot8siQQhCik4BOEiZpYAjXRFERaagL+K8snEYuiVs+yMNJzs6OQqaTzrkN69JxP7wcrOBMJCmqVo8M7Ps7jw0BTAGcvro5qHWikErPPPKB2AmcvrbcW9sJRClvKtKECaGnal8ZRPv1kSCUIUUnAIwkVYb7uKAigWQiQZsBOLIaOcGa1VADCxequt2I8xhX2432VnpAmd3+41KinIQbUNhUB//pMdp3HT73ZElCStEnJ2RrplBWSeEspSvlUVeHxuKfr0MLcm+s2SSBAikIJDEC7Ce9tNBeHgRSwGK8tJO1b9gWO2z5cfyGJmLgHAjCfquRWE3bKW2VUI8gNZqH3vKHPcnSpw+7pdptYcMyWU52rSCk0GQ+2m7jA/WRIJQgRScAjCZcw6eSczbsdiWLm7nJ5Pcxd984n6qM9VhK0hZpYgN2q+2K1JU7UhVrnRY2bJMlNCy4f0DccI6dZcsyxRGjiRirBttQRBOCI/kIXyIX2TXqnRo1mn0hUFgDN3G8/SEAy1u3q+Ex1nmJ9rsTgsahqaMbF6K+aufAMTq7eipqFZ+HxOEYnhAs4pLUY0pVCPUSnU3F7af0WuBUEkI2TBIQhCGLdiMUTdXU7PV5TbIyZTCAhbKViWoETXfGHGcJ39r0jtJLOAYKN1SEV4bo986xJKAydSElJwCIKQwo1YDBH3k95NVD6kr+2xVhticZSzLhjWHBJd84WnoAAQDlrmKYW8uUGNTRdXOAogQSQTpOAQBMHEy95DVqnHbsaEaAJ/x8FWKApMu7f7oeYLT0GRsWSxlFDe3Ar6MI4l4CYjCL9DCg5BEDHEI+iUJ8i9cBPlB7JwTYl4sb5E13xhKShOLWe8uZ3oOBOjz6iQ73JOEH6DFByCSHLMLC12rDDxjENhCe1Eu4lk437cXn8v4fVNS7TViiC8gBQcgkhizCwtdq0wThQMNwQ6r6fTniPHbcfiyCJqLfFi/b3GODe/WK0Iwm0UlVfTPIVpa2tDIBBAKBRCr169Ej0cgrBFMNQeVekXCL9511VOAQDudyJKip193RToK149EG6OqUN0/PHCq/X3Gp4SGgy1J30xSiL1kZHfVAeHIJIUM0uL2XdW2Kk/s/twKypdrKUy8sJAzGes8WvVd63OI7qdFfrjeLX+XmJW4ycVazcRXRtyURFEkmKV8eO0CrBoHEpNQzOztYCTuBmRbCZRi5FbliXjce6dVuzZ+ntBomv8EES8IQsOQSQpZpYWN6oAi7zRm7UWcCLQ8wNZmFk6MOqzGaUDLDOtjBYat6r0so6zbPN+3FtR7Nn6u41frUoE4RVkwSGIJMbM0hKPDtC81gL6Pkd2CIbasbHxSNRnzzd+hHuuGob8QJZwILRbGVm844wa2Bt1lVMStv4y+KHGD0HEE1JwCCLJMcv48boDNEtopgHYuGgCSgpybB/XSjERFdZuCXWz41itvzYf/d+JgLKliK4GuagIgrANyxWzZNZIR8oNYN00UtQF5JaryO5xEtm4k8XssYNQVzkF6xaOR13lFF+krROEV1CaOKWJE10UN4vQeZFiXNPQHGNtMApk0fOabaetQ4+MdJzoOGO6HjLztEq391sRQIJIBmTkN7moCKIL4nYROi9cYSIxLKLn5W2nXwcNs/UwHsdMSTFzs9W+d9SXRQAJIpUgFxVBdDHcyizyYlzGWjVe1mYxroOG6HpYuZ94brbsjLSY9a/asDfh608QqQYpOATRxfBjunAiYlV4GWCA9XqIKIm8uJ3Dre1MpWrNa02O5kMQRDTkoiKILobf0oUTVYCO1/MKsF4P0fRzo5ut9r2juH1tI/OYq2qbMH9iEcXjEIRLkAWHILoYfitCZ6YsuNViwYgWO3PvtHOF+jRE1sMqy0uP5mYDwC2KCACdABXdIwgXIQsOQXRB/FSEjmdR2vPhcdyw6nVmIK5ZcK9ZM8mmYyew98MQlm7ed67lQkUxRg3sjeyMNJzs6BRaDzs1ZcxcYtqcqegeQbgHpYlTmjhBJBxjSvjiimFYumkfM8XaLAOJlx3GypYyHteOkuc0bVw/BlYaPEEQ0fimm/jPf/5zTJgwAdnZ2ejdu7fQPqqq4oEHHsCAAQOQlZWFK664An//+9+jtjl16hRuv/125ObmokePHrj22mvx4YcfejADgiDigbEA3ciBAabbauehVm5wLy+WZ/fhVq5yox3XrmtIJsuL5RqsqiimonsE4RGeuqg6Ojrw7//+7ygvL8fq1auF9lm2bBl++ctf4umnn8bQoUPxs5/9DN/4xjewf/9+9OzZEwBw55134oUXXsBzzz2Hvn374u6778Y111yDHTt2ID093cspEQThEcYaMyy3VaeqcuN1VLC/azjY6hvXkJ9cgwSR6nhqwXnwwQfxwx/+ECNHjhTaXlVVPPzww/jP//xPXHfddRgxYgR++9vf4uTJk1i7di0AIBQKYfXq1XjooYfw9a9/HaWlpXjmmWewd+9e/PWvf/VyOgRBxAleIPSYwj7c4F5e4O/YwpyYz/XfxzvA2svaPgRBnMNXWVRNTU1oaWnB1KlTI59lZmbi8ssvR319PQBgx44dOH36dNQ2AwYMwIgRIyLbEASR/LD6JpllgPG+KynIiXUNXU2uIYJIdXyVRdXS0gIA6N+/f9Tn/fv3x6FDhyLbZGRkICcnJ2YbbX8jp06dwqlTpyJ/t7W1uTlsgiA8gtViwczNw/uOXEME0fWQtuA88MADUBTF9N9bb73laFCKoS6Fqqoxnxkx22bJkiUIBAKRfwUFBY7GRxBEYjFz8/C+I9cQQXQtpC04t912G+bMmWO6TWFhoa3B5OXlAQhbafLz8yOff/zxxxGrTl5eHjo6OtDa2hplxfn4448xYcIE5nGrqqpw1113Rf5ua2sjJYcgCIIgUhhpBSc3Nxe5ublejAVFRUXIy8vDli1bUFpaCiCcifXqq69i6dKlAICysjJ069YNW7ZswfXXXw8ACAaDePvtt7Fs2TLmcTMzM5GZmenJmAmCIAiC8B+exuA0Nzfj008/RXNzM86cOYNdu3YBAP7lX/4F559/PgCguLgYS5YswcyZM6EoCu6880784he/wMUXX4yLL74Yv/jFL5CdnY25c+cCAAKBABYsWIC7774bffv2RZ8+fXDPPfdg5MiR+PrXv+7ldAiCIAiCSBI8VXD+67/+C7/97W8jf2tWmZdffhlXXHEFAGD//v0IhUKRbRYvXoz29nYsWrQIra2tGDduHP7yl79EauAAwK9+9Sucd955uP7669He3o4rr7wSTz/9NNXAIQiCIAgCALVqoFYNBEEQBJEk+KZVA0EQBEEQRCIgBYcgCIIgiJSDFByCIAiCIFIOUnAIgiAIgkg5SMEhCIIgCCLlIAWHIAiCIIiUw1fNNuOFlhlPTTcJgiAIInnQ5LZIhZsuqeB89tlnAED9qAiCIAgiCfnss88QCARMt+mShf46Ozvx0UcfoWfPnpZdyolotEalhw8fpiKJLkFr6g20ru5Da+o+tKZyqKqKzz77DAMGDEBamnmUTZe04KSlpeHCCy9M9DCSml69etGP0WVoTb2B1tV9aE3dh9ZUHCvLjQYFGRMEQRAEkXKQgkMQBEEQRMpBCg4hRWZmJn784x8jMzMz0UNJGWhNvYHW1X1oTd2H1tQ7umSQMUEQBEEQqQ1ZcAiCIAiCSDlIwSEIgiAIIuUgBYcgCIIgiJSDFByCIAiCIFIOUnAIS37+859jwoQJyM7ORu/evYX2UVUVDzzwAAYMGICsrCxcccUV+Pvf/+7tQJOI1tZWzJs3D4FAAIFAAPPmzcPx48dN9/nud78LRVGi/o0fPz4+A/Yhy5cvR1FREbp3746ysjJs27bNdPtXX30VZWVl6N69Oy666CI8+eSTcRppciGzrq+88krMPakoCvbt2xfHEfub2tpa/Ou//isGDBgARVHw/PPPW+5D96o7kIJDWNLR0YF///d/x3/8x38I77Ns2TL88pe/xOOPP46Ghgbk5eXhG9/4RqQPWFdn7ty52LVrFzZv3ozNmzdj165dmDdvnuV+06ZNQzAYjPx76aWX4jBa/1FTU4M777wT//mf/4nGxkZcdtllqKioQHNzM3P7pqYmXH311bjsssvQ2NiI++67D3fccQfWr18f55H7G9l11di/f3/UfXnxxRfHacT+58SJEygpKcHjjz8utD3dqy6iEoQga9asUQOBgOV2nZ2dal5enlpdXR357IsvvlADgYD65JNPejjC5OCdd95RAaivv/565LPt27erANR9+/Zx97vxxhvVb37zm3EYof+59NJL1VtvvTXqs+LiYrWyspK5/eLFi9Xi4uKoz2655RZ1/Pjxno0xGZFd15dfflkFoLa2tsZhdMkPAHXjxo2m29C96h5kwSFcp6mpCS0tLZg6dWrks8zMTFx++eWor69P4Mj8wfbt2xEIBDBu3LjIZ+PHj0cgELBcn1deeQUXXHABhg4dioULF+Ljjz/2eri+o6OjAzt27Ii6vwBg6tSp3PXbvn17zPZXXXUV3nrrLZw+fdqzsSYTdtZVo7S0FPn5+bjyyivx8ssveznMlIfuVfcgBYdwnZaWFgBA//79oz7v379/5LuuTEtLCy644IKYzy+44ALT9amoqMCzzz6LrVu34qGHHkJDQwO+9rWv4dSpU14O13ccO3YMZ86ckbq/WlpamNt/+eWXOHbsmGdjTSbsrGt+fj6eeuoprF+/Hhs2bMCwYcNw5ZVXora2Nh5DTknoXnWPLtlNnAAeeOABPPjgg6bbNDQ0YMyYMbbPoShK1N+qqsZ8lkqIrikQuzaA9frMnj078v8jRozAmDFjMHjwYLz44ou47rrrbI46eZG9v1jbsz7v6sis67BhwzBs2LDI3+Xl5Th8+DD+53/+B5MnT/Z0nKkM3avuQApOF+W2227DnDlzTLcpLCy0dey8vDwA4TeR/Pz8yOcff/xxzJtJKiG6pnv27ME///nPmO+OHj0qtT75+fkYPHgw3n//femxJjO5ublIT0+PsSqY3V95eXnM7c877zz07dvXs7EmE3bWlcX48ePxzDPPuD28LgPdq+5BCk4XJTc3F7m5uZ4cu6ioCHl5ediyZQtKS0sBhP37r776KpYuXerJOf2A6JqWl5cjFArhzTffxKWXXgoAeOONNxAKhTBhwgTh833yySc4fPhwlBLZFcjIyEBZWRm2bNmCmTNnRj7fsmULvvnNbzL3KS8vxwsvvBD12V/+8heMGTMG3bp183S8yYKddWXR2NjY5e5JN6F71UUSGeFMJAeHDh1SGxsb1QcffFA9//zz1cbGRrWxsVH97LPPItsMGzZM3bBhQ+Tv6upqNRAIqBs2bFD37t2rfutb31Lz8/PVtra2REzBd0ybNk0dNWqUun37dnX79u3qyJEj1WuuuSZqG/2afvbZZ+rdd9+t1tfXq01NTerLL7+slpeXqwMHDuySa/rcc8+p3bp1U1evXq2+88476p133qn26NFDPXjwoKqqqlpZWanOmzcvsv0HH3ygZmdnqz/84Q/Vd955R129erXarVs39f/9v/+XqCn4Etl1/dWvfqVu3LhRfe+999S3335braysVAGo69evT9QUfMdnn30WeWYCUH/5y1+qjY2N6qFDh1RVpXvVS0jBISy58cYbVQAx/15++eXINgDUNWvWRP7u7OxUf/zjH6t5eXlqZmamOnnyZHXv3r3xH7xP+eSTT9QbbrhB7dmzp9qzZ0/1hhtuiEm11a/pyZMn1alTp6r9+vVTu3Xrpg4aNEi98cYb1ebm5vgP3ic88cQT6uDBg9WMjAx19OjR6quvvhr57sYbb1Qvv/zyqO1feeUVtbS0VM3IyFALCwvVX//613EecXIgs65Lly5VhwwZonbv3l3NyclRJ02apL744osJGLV/0VLpjf9uvPFGVVXpXvUSRVXPRi8RBEEQBEGkCJQmThAEQRBEykEKDkEQBEEQKQcpOARBEARBpByk4BAEQRAEkXKQgkMQBEEQRMpBCg5BEARBECkHKTgEQRAEQaQcpOAQBEEQBJFykIJDEARBEETKQQoOQRAEQRApByk4BEEQBEGkHKTgEARBEASRcvz/Mea7PCN6Q6wAAAAASUVORK5CYII=", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "arrows = [random_arrow() for _ in range(1000)]\n", + "plt.axis('equal')\n", + "plt.plot([a.real for a in arrows], [a.imag for a in arrows], '.');" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "A simulated tournament consists of placing random arrows until one of them is further away from the center than a previous one:" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [], + "source": [ + "def simulate_archery(previous_arrows=()) -> int:\n", + " \"\"\"Simulate the archery tournament and return the number of arrows fired.\"\"\"\n", + " arrow = random_arrow()\n", + " if any(abs(arrow) > distance(a) for a in previous_arrows):\n", + " return len(previous_arrows) + 1\n", + " else:\n", + " return simulate_archery([*previous_arrows, arrow])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now we can run this for say, a million times and take the mean:" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "2.718797" + ] + }, + "execution_count": 25, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "mean(simulate_archery() for _ in range(1_000_000))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Wow! 2.718! I know that number, it's a [**friend from work**](https://www.youtube.com/watch?v=wjwD1ZGz1UM)!\n", + "\n", + "I know that *e* = 1/0! + 1/1! + 1/2! + 1/3! + ...\n", + "\n", + "Now I recognize how this applies to the problem! Before seeing this number, if you had asked me, I would have said that the average should be somewhere around 3, and if you told me it was a famous irrational number, I would have guessed π, not *e*. But upon seeing the factorial signs, I see that it doesn't matter that the arrows are on a circle, what matters is the relative distances of the arrows, and for *n* arrows there are *n*! possible orderings. I wrote up how I went from that insight to a proof that *e* is the answer, but unfortunately I can't find the writeup now, so I'll repeat [the solution given by 538](https://fivethirtyeight.com/features/can-you-hunt-for-the-mysterious-numbers/):\n", + "\n", + ">Solver Balthazar Potet approached this by thinking about the values for the first N arrows Robin fired and the probability they’d result in a score of N. With any N values, there were N! ways to order them. For Robin to have a score of N, the smallest value couldn’t have been in the Nth position, since it had to be greater than the previous value. And when each of the other N−1 values occurred in the Nth position, there was exactly one way to order the remaining values so that they formed a decreasing sequence. So of the N! orderings, N−1 resulted in a score of N, meaning the probability was (N−1)/N!\n", + "\n", + ">From there, you had to use these probabilities to compute an average score, which you could find by multiplying each score by its probability and then adding up all those products. The probability Robin scored 2 was (2−1)/2!, or 1/2, which meant a score of 2 contributed 2·1/2, or 1, to her average score. The probability Robin scored 3 was (3−1)/3!, or 1/3, which meant a score of 3 contributed 3·1/3, or 1 (again!), to her average score. In general, the probability Robin scored N was (N−1)/N!, which meant a score of N contributed N·(N−1)/N!, or 1/(N−2)!, to her average score. Since N was at least 2 — meaning Robin fired at least two arrows — her average score was 1/0! + 1/1! + 1/2! + 1/3! + …, a sum that converges to *e*, which is approximately 2.71828. Huzzah, another riddle whose answer was a famous mathematical constant!" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] } ], "metadata": { "kernelspec": { - "display_name": "Python 3", + "display_name": "Python [conda env:base] *", "language": "python", - "name": "python3" + "name": "conda-base-py" }, "language_info": { "codemirror_mode": { @@ -1067,9 +997,9 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.3" + "version": "3.13.9" } }, "nbformat": 4, - "nbformat_minor": 1 + "nbformat_minor": 4 } diff --git a/ipynb/houses4.jpg b/ipynb/houses4.jpg new file mode 100644 index 0000000000000000000000000000000000000000..12d48d29430f12c2dc301b259bf7cafe9d21422a GIT binary patch literal 122667 zcma%ibzGD|_wP~?3JB6Euyje+(y*j7OCzv=bT=4)ghkv#0H~;_Xb39*`$56P!gzp+hK`M>_IZq`MnOYCj*a((_yOue6hsXM-~j<4 zCe{-=B6=QaY+_zbb61AIxT+2kAR`l>j26VrGd{bfmJFQc3~TVoW4 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