158 lines
42 KiB
HTML
158 lines
42 KiB
HTML
<!DOCTYPE html>
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<html lang="ko">
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<head>
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<meta charset="UTF-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
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<title>BOJ 30524 - Offline</title>
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<style>
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:root {
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--bg: #fafaf8;
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--paper: #ffffff;
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--ink: #1e1f24;
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--muted: #6a6d75;
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--line: #d8dce3;
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--accent: #0d6e6e;
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--code-bg: #f4f6fb;
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}
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* { box-sizing: border-box; }
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body {
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margin: 0;
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background:
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radial-gradient(circle at 15% 0%, #f0efe9 0%, transparent 42%),
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radial-gradient(circle at 85% 20%, #e7f1f2 0%, transparent 38%),
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var(--bg);
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color: var(--ink);
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font-family: "Noto Sans KR", "Pretendard", "Apple SD Gothic Neo", sans-serif;
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line-height: 1.65;
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}
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main {
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max-width: 980px;
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margin: 0 auto;
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padding: 24px 16px 56px;
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}
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.header {
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background: var(--paper);
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border: 1px solid var(--line);
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border-radius: 14px;
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padding: 18px 20px;
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margin-bottom: 18px;
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}
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.header h1 { margin: 0 0 6px; font-size: 1.5rem; }
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.header p { margin: 0; color: var(--muted); font-size: 0.95rem; }
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.header a { color: var(--accent); text-decoration: none; }
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.section {
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background: var(--paper);
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border: 1px solid var(--line);
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border-radius: 14px;
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padding: 16px 18px;
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margin-bottom: 14px;
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overflow-x: auto;
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}
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h2 {
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margin: 0 0 10px;
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font-size: 1.05rem;
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color: var(--accent);
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border-bottom: 1px solid var(--line);
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padding-bottom: 8px;
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}
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pre, code {
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font-family: "JetBrains Mono", "Fira Code", monospace;
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background: var(--code-bg);
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}
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pre {
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padding: 12px;
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border-radius: 10px;
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border: 1px solid #e7ebf2;
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overflow: auto;
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}
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blockquote {
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margin: 14px 0;
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padding: 16px 16px 14px 22px;
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border-left: 4px solid var(--accent);
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border-radius: 10px;
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background: linear-gradient(90deg, #eef8f8 0%, #f9fdfd 100%);
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color: #24313a;
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font-weight: 600;
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position: relative;
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}
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blockquote::before {
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content: "“";
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position: absolute;
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left: 8px;
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top: 2px;
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font-size: 1.35rem;
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line-height: 1;
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color: #0b5f5f;
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opacity: 0.7;
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}
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blockquote > :first-child { margin-top: 0; }
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blockquote > :last-child { margin-bottom: 0; }
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q {
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color: #114f50;
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font-weight: 700;
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background: #edf8f8;
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border-radius: 6px;
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padding: 0 4px;
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}
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.math-inline math {
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font-size: 1em;
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vertical-align: middle;
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}
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.math-block {
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margin: 10px 0;
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padding: 8px 10px;
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overflow-x: auto;
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background: #f8fbff;
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border: 1px solid #e2ecf8;
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border-radius: 8px;
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}
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.math-block math {
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font-size: 1.04em;
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display: block;
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}
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table { border-collapse: collapse; width: 100%; }
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th, td { border: 1px solid var(--line); padding: 6px 8px; }
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img { max-width: 100%; height: auto; }
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</style>
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</head>
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<body>
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<main>
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<header class="header">
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<h1>A Pivotal Question</h1>
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</header>
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<article class="section">
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<h2>문제</h2>
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<p>Quicksort is a recursive sorting algorithm developed in 1959 by Tony Hoare. One of the major steps in the algorithm is the <em>partition\/</em> step: given an element <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>p</mi></mrow></math></span> in the array (the <em>pivot\/</em> element) rearrange the elements in the array as shown below where all the values in <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mi>X</mi><mi>L</mi></msub></mrow></math></span> are <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo>≤</mo><mi>p</mi></mrow></math></span> and all elements in <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mi>X</mi><mi>R</mi></msub></mrow></math></span> are <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo>></mo><mi>p</mi></mrow></math></span>.</p>
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<p style="text-align: center;"><img alt="" src="data:image/png;base64,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" style="width: 280px; height: 41px;" /></p>
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<p>Figure A.1 below shows an array before and after it's been partitioned with the pivot element <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>13</mn></mrow></math></span>. Note that the elements in <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mi>X</mi><mi>L</mi></msub></mrow></math></span> and <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mi>X</mi><mi>R</mi></msub></mrow></math></span> are typically not in sorted order and either one of them could be empty.</p>
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<p style="text-align: center;"><img alt="" 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q/CbU05vTrmJm02vHDNU4RZiTx5jMgFqZdbfZQ6w4hxpxJKQtB4pUR6SMjLWQhr78x7i93SPlKARGRnkwt72f/koXMBxyH2ozDj8l1DLLaTWtxxRJSlJaTMzPQRCuUy/rWqdQYhQ61FXJkaGEqxQT/8ADUoiJf8AZMwFnABmGVvzyyde9f8AdA08BXrkvS3LZnw4Vw1iJTX5aFuM+ErzEqSkyJXjn4peUWgzIz04ajHURlJsha0pReFvKUo8CIqiziZ//wBAJK4rqoduKYTWqkxFdfx3lo8VOOYa81CSNR4eoh2qDW6ZcFORPos5idDWZpJ1lZKLEtZH6DL0HpASAy7Jbyk5S/bI/wBzgDURFVy4aRQlRE1eoxojkt5DEdDi8FOuKPBKUp1npP7OcBKiKeuGkM1+NRHKjGKryEqW3EJeLhpSWJmZFqLD04YgJUdepfo6V/CX9xgIPJnyb2p7pifJQLIAAAAACpQ+Vmre5Ifz5QC2gAirhuGkW5ETJrlRjQWVqJCDeXga1GeBEktZnp5hKgAAMu3OHJq37Y995DUQEU9cNIZr8aiOVGMVXkJUtuIS8XDSksTMyLUWHpwxEqAiqbcNIqdWnUynVGNKnQSQqS0yvONrOxzc7DQR+KejWJUBmG6F8zad71Y+5Y08AEfXK1TaDC8LrE1iHHNRIJbqsM5R6kpLWZn6C0gOpb110O4Xn2aPUmZEhgiN1jSh1sj1GpCiJRF68MBNgAy+jfrD173M38TYDUBGXFX6VbdNcqFdqEeBDRrcfWSSP1EWsz9RaQEk2tLiErQZGlREZGXOQq1Vyh2nSZb8afXIrbsdWY/hitLCvQ4pJGlB+pRkAssSSxMitSYb7UiO6kltutLJSFpPUZGWgyHXrv6EqHs7nwmApG575IKB/eP5hwX6bKZgw35ctwmo7DanXVnqSlJYmf2EQCq/+p1jcHlN43UPeDLH89bz/ozMc7H1YYixzKrAg0pVTnS2YtPQ2Tqn31b2hKT1GZqww1lrAQ1Hvy2KvU26dArEdU5xOe2w5nNLdL0oJZFnF9GIswDKcu35/k5+tUL4jGrAIy4K/SrehplVufHhMKUSEKdVga1HqSktaj9RYmOpa94W9dW/lb9WizXGDwdaQrBxv+sg8FF9pAOW4LootvuR2qvPaYfkY7ywRGt13DWaUJI1GRc5kWgdig1ymXBAKbRZzE2NnGg1sqxzVFrSotZKLnI8DASIyXJxy85W/wDCv5dQDWhE3BcdHt1tldaqMeHvyjSyhxXjumWskJLxlH6iIwHLQa5TK/DVKo8xqWylZtrNB6ULLWlRHpSerQZEYg52Ue0IEp1iVX4TamnN6dcxM2m144ZqnCLMSePMZkAtTLrb7KHWHEONOJJSFoPFKiPSRkZayFbypcmV3e55nyVgODJFyX2r7uY+AhbVrS2hS1qJKEliajPAiL0gKvTcoVp1OpswINehPSX1GhkkqPNeUWskLMs1Z+pJmJ+qVGFSae9OqcpiJDZTnOPPLJCEF6zMBE29elu3FMciUeqx5MtCN8NjShZo/fJKiIzT6y0CwAMtyocqWTH2mX8LY1IBW6/fNsW7V2qZXq3Cps11nwhCJTm9pUjONOOcfi6yPRjjoHFGyiWXKkNR4120B191ZIbbRUGlKWozwIiIlaTM+YB27iu+gW4+0xWaoxGkupNaGNK3FJLWrMSRqw9eGAkaNVYFbprNQpExibCeLFt5hZKSr06S+4B3Rl+R/wA68onvhX3rAagKvWcoFqUaoOwqnXIbElnDfkGo1Ezjq3wyIyRj/SMgFmZdbeaQ6ytLjS0kpC0HiSiPUZHzkI64a9TLdgomVqWiLHW83HQpRGec4tWalJERGZmZn/lieojASYjLp82av7G98BgOGyvM2g+wR/lpEyACpI5WXfciPnqAW0Y7ulrZojmTG5667SoTlYTHaSmYtolOoInUF4qj0lo0aAGiZP8AzDtv3bG+UkUqkQKNkoql31q5rwaNivS1T24ss0tm3hnGaW05xm4eCkp8Ui0JSWACK3Ltu1Sj2rWKhU4jtPj1ieqZDgOFgbLRloPN/Zx9HoSR842gBl+5781Kx74kfcgWfKrUZlJya3PPpalomx6e840tHlIUSD8cvWnX9gCm5EqNTJG55psOQ00qJPhPrlmoiPfFLUvPUo+c+bH1F6B1dypcLtVyNRjqD+JUt92HvritTaSStOJnzJSsi+giAU3KW/LuC9cmt0SFut06TckZilRjxIvBicSZvqL950yIy9CCRzmY9LgMt3SfJbK9pZ+IakAwvLw85NyqZLKFMIzosioKkOtK8h5xCkZpKLnwx1f0zH7uop6bcTY9zxTJup0+soShZaFKZUhRuIM/3TzSIy9YD8iPOVbddTo9UI1sUeib5TkL8lKl71nLSXpPfHCx16PUP27Z6bf3U9qKhmTfDdMVGnITo30iNze1K9JkaUlj6E4ANzGVRf1nZf1W/wCygBqoAPP26jtmiQLViViHSoTVVlVuNv0xLRb6vEl4kateGgtA9AgA+HvyLn9UwGV7lnkLtz6ZP8y6NOqn6MmfwV/CYDznk9pOSmlWfb1xVO2ZClohRjmVV6FIeiIkGhJKM87FHlmZGok5pHzlgNgyn2knKPYx0mFWTgMyFsyES2Ek6laUmSi0EosSPQZGR6yIwGVZdnY1+RbUsmzagzWrhZmokPTIzpOHDabQaFOurRoRipSTwx1lqxwHolJYJIjMzMi1nzgMq3Qf5jY31qhfC6NWAZzeVtXJUsq1nVqmuQVUKmodKS3I0qQpeg1oTh5RpwIj1lp5j00XLiRR8ueS52gIwuF2SaZO9aFLi56C8fD9nN377CV6AExY8rjBuj75lyizyocNinwyVp3pK/GWZegzUlWn0HgFRlcA7qilMRC3tm4qKpEpCdBLdaNxSXDLnUSWyTj6DMBtAy2zuXrKB7PC+SgBqQ86ZJbyi16+rvumdCn1erKlqgwY8JjfjhQmyLxixMiQS84ufFRpVgR6QGg5MaPbcqlXVUbSrNQkHXpby5bkhR77EfMjI0Eg0kaDTnaladWkywFOrbFHya5EqpYkyvNV+tyI70WJBIi8IccfxJtKGSUpRERqJRH/AJcwDQsiFv1K18llApFbUfCDDSlOINWdvectSyRj/RJRF9mjQJ2+/Me4vd0j5SgERkZ5MLe9n/5KFzAZzl+ZqrmTp56iwjqC4kuPLkQSIz8JYbcJS2zItZHgWJc5EYoVwZSaHlesZu3rWjSF3TPdb3mI43mnT1IWlSpCnNRJQWOBkeJ4kWGkyIPQSSMkkSjzjItJ+kfoDMMrfnlk696/7oGngKTc+TiiXZdaarc0RipRmYSI0aK6k8G1561LXoPTiRoL1Zp+kUOgZOrOreUhcihUCBFo1sv72462g8Zc8iI8zSZ+I0RkZ+lZkWpOkPnI485V8u2VOoVUjXOgvNQYpr1tx85wsE+gjJtB6PT6x+2nPTR91Pd1EhmSIVUp7c11lGhJSUpQedh6TJSjP0meIDcxl2S3lJyl+2R/ucAaiPP2X+2aJSKtY9QplKhRZ8y6Y65EltoicdNSjUecrWZY6cNQDfpDLciO6w+gltOJNC0nqUkywMhgsi36Rbu6gs+LQabEp8dVGkLUiM0SCUr8L4x4az9ZgN9HXqX6Olfwl/cYCDyZ8m9qe6YnyUCyAON95phBKfdQ2kzwxWoiLH7Rw8Iwf/Mjf/Kn/wDQDhGD/wCZG/8AlT/+iiqyyWYxeU+2alU00+fEWlBOysEsO4pJWKXMTSXlYeNhpAXqFUIU9BLgzI8lKiziNl1KyMvToPUK5D5Wat7kh/PlALaADz9uo7ZokC1YlYh0qE1VZVbjb9MS0W+rxJeJGrXhoLQPQIAADLtzhyat+2PfeQ06Qy3IjusPoJbTiTQtJ6lJMsDIBgsi36Rbu6gs+LQabEp8dVGkLUiM0SCUr8L4x4az9Zjb63SYFcprtPq8RqZBew3xl1OKVYGRliX0kRgMayM0+HS8uOVKFTIrEOGz4ElthhskIQWYo8CSWgtI3MBmG6F8zad71Y+5Y08AGLZarkl2Jf1rXZOpb1StqPHfiPm0WKojrik/hSx0Eo0lmljhiWcWJYgOemV+nZSMp1sVqzM9+BRWXzqNSNs2yMnmsERtJEalZxksy1Jw14mNiABl9G/WHr3uZv4mwGoDCN1ZbNEayZ1yuopUIq069GJU02iN7DfEJwJWstBYaOYBttNLGlxSIzLFlGkubxSGNMxKLkgyX163K3czVbnTCkOx4T2CX3zeTmpaQ1nKUZKVjierFSj0ALJudbaqtqZKqZT68S25qluP+DrPSwlasSQfoPnMuY1GQv1d/QlQ9nc+EwFI3PfJBQP7x/MOCy5QfMK5fdkn5SgGN2dTsktvU+j3A7bj0ZC0M4ViXDkLiE8ZERKJS8UpLO1LwJOOoxpWVqyV5QLcgwY9cdpBRprU8pLKM/OzCVgXlFh5RKI8dBkR4GAzXLA/GykXvZFFseWxUptNm+HTKjDcJxEFojT5TicUkZ5pmScccUlo0kPQYDKcu35/k5+tUL4jGrAM5rFs3JJy40K4mHIK7biU1yM629pcbcUajNTZYaFH+CLOI/JSZH66JlCIo26ksVVvIwqr8ZRVMmtGfH8YsXMNeCSVr/dR6CATOSGVxhyyZUKxLLPfgSWqTFNWneWmzcJSU+glKQSj9YUiVwHuo6zSoxZkSuUdua62nQk30HmkvDViaSVj6ccQG0DJcnHLzlb/AMK/l1ANaHnTJLeUWvX1d90zoU+r1ZUtUGDHhMb8cKE2ReMWJkSCXnFz4qNKsCPSA0HJjR7blUq6qjaVZqEg69LeXLckKPfYj5kZGgkGkjQac7UrTq0mWAp1bYo+TXIlVLEmV5qv1uRHeixIJEXhDjj+JNpQySlKIiNRKI/8uYBoWRC36la+SygUitqPhBhpSnEGrO3vOWpZIx/okoi+zRoEllS5Mru9zzPkrAcGSLkvtX3cx8BCi7ripTqdkffRAW4hMyY1GkKQeH4IyUoyM/QZpSR/ThzgO9l7pFOi5AqnHjttssUyPHXCUjRvKkLQSDQZajw0Yl+8fpGf3bX5V3VPIUxVjNdOqZtzJaFfk5EhBNl4xc+BmrR/TAWfdRT024mx7nimTdTp9ZQlCy0KUypCjcQZ/unmkRl6xuYDLcqHKlkx9pl/C2NSAUWu5MqBc12zK1dMGPVc6MzFitPJPCOlBuKVqPTnGv8A0FLsWwLSq2UF24KFQ4UOi0B5cSGppJ/jktJlvjx4npQ2fip9Ks4+YgHDud3nKtfOU+sVQjXVuFfBDNzSpplBrJLZegiwIvXml6B+5KZ6abugspVsQjIqa5vdSQ0nyW3TS3vhkXMZm7p/qkA3MZfkf868onvhX3rAXm8ZkmnWjXJsAjVMjQX3mSIsTNaW1Gn/AFIhmm5ggxJeRKG7JbRIdqr0tyebvjG+o3VoPPx14pSRaQHS3JtYXMsOqUtbxvM0ipux4y1HjgwZEpJY/SavswFVy7SpFzR7duNLq00Nq4o0GltkeCZCfHN2SZc5KUgko/opM/2wHpkRl0+bNX9je+AwHDZXmbQfYI/y0iZABUkcrLvuRHz1ALaM03SXIjdP8Fr5zYC25P8AzDtv3bG+UkVrJrecXKlR6s7NtxyLDjSlxEompJ1EhJaz0pIseY06cPSYCg7mOfLaunKHb0V1x62qXUVJp+crOSyRuuJzEn6DSlJ4atGPOY9AgMv3PfmpWPfEj7kDR6ouK1TZblR3vwFDK1P74WKd7JJ52Jc5YYgMZodMoLWSybULYu2oIyekiRIVCW0klEgjUbjCHFpJxCFKJRYaVHnHgrTiIDJRZdyNZBKRT6fDiOJq8459QjSZSo5vRFEWDeeSF4Z5IRjo8gzLRjoD8y2VC6DuPJm1ULepMNxFeYXERHqinUuLSpBEhRmwnMTpLxiJWHoHoOkOznqcy5VorEScoj3xlh83kI0nhgs0pM9GB+SX+4DOd0nyWyvaWfiGpAM7yxNWlLZt+Fdc9+nT36g3wRLjJPfmZJGWapJklREWJpxzizdJeoyzLLzacytVyyaDOrL9arM+pEbTZtIZRHiISZvLNCOc8Umaj/cwIi04hot4RLaqOVSix49ZfpN/Mw1uxnI7WeTsY87FDpKSaFJ0LMixJRYYkZChFbLs/dQ0VS6g9V5VHgKm1SW6SUE2pRLSy0lCdCCLFCiTrPFRmZniA9EjKov6zsv6rf8AZQA1UAGK7rLk7pXvuL9yxtQAPh78i5/VMBle5Z5C7c+mT/Mui65Ra8/a9j1qtxIaZr0GOp4mFKwJRFrxP0EWJn9ACBuOYT2QqqSq3T2aXvlCdN6EkyNDBmyZZicNGvAiIvUM4sa/puTnc/2LKqFGm1aVUJCojDDas1eYpxxTeGg8TNBFmpw0lhpIBI7q+PEpliQbigkmBX4VQa8DmMkSHiMyVnJzi0mWBY4f0RsdrSpc62aRLqTe9Tn4bLshvDDNcUgjUWHqMzAZ5ug/zGxvrVC+F0asArd+XdBs6jFMmYOyX1kxDiktKFSXj1JIz0EXOaj0JLEzGfWwmi25U1XTcNXp1dvetyo8PNhyELKOl1xLaWWE44khBKxUrWZJMwHXybMKpO6NykwpBZi57Eaexj/7jeozL1EpeH0kFfjqqu6xtk42C00ehuyJBl+xn762RH6/wiT+0Btoy2zuXrKB7PC+SgBqQwbcnUxFEp160l9BIqcKtrZfIy05hJIkfZiSzL6QEZk+rS7bvrLhX2Y7smjwHjfW01oJx9G+GoiPUR+VnHzYkYv0qbS8pGRGVW67RSgsy6c+/vUkiUtjNJWa4lZkX7pKSrAtGACP3K9aq1cySRXa2468uPJcjR3nTxU4ynNwxPnwM1J/sjQr78x7i93SPlKARGRnkwt72f8A5KFzAZzloqUxiPbFJjzXadErdXagTJjKsxaGjSpRoSr9lS80kkrm0ilZaMj9uQLXlXTaqU23WqHHOUy9EVvaHN7LEiUX754YEotJmenEBs9qy5c+16PMqTe9TpENl2QjDDNcUgjUWHqMzEoAzDK355ZOvev+6Bp4CAygVxVtWPXqy3hvsKE682RliRrJJ5pH/awFW3OcfeMjVuLXip6Q25JdWrSpa1urUajPnPSA6rsGgVPKzWVWnX36TekeIhFTbajk4y82ZJNBrStOapSSNGBpURljgeOkhT8lduk7uirwrDEl6oRqZFTCfnvmRqfmLzDXqIiLNJK05pERJIkkA9BDLslvKTlL9sj/AHOANRGK7pX8pk6+s0X/AHAbUMVuj9ayzfcj/wD9wDah16l+jpX8Jf3GAg8mfJvanumJ8lAsgDpValU6sRkx6vAiT46VEsmpTKXUkoiMiMiURljgZ6fWYiuIto7LUHq9nsgHEW0dlqD1ez2Rj9V3NlKr2UKq1qqTEQ6K84hUem01omsEkhJHnKwwTiZHoSXPrIBq9k5PbWshKztmkMQ3VozFv4qcdWnEjwNajM8MSI8NWgh8w+Vmre5Ifz5QC2gAxXdZcndK99xfuWNqAAAZducOTVv2x77yGogMVuj9ayzfcj//ANw2oBiuSvl/ysfTC+WY2oBmG6F8zad71Y+5Y08AGWV5tF15ZytivqJdCgUlFRbpq/yc51Tppz3C/bQjNLBJ6M48TAVGvZP42TbKnaNesR1yGzWammnTqSSzNtba0qUtaS15qSSZ4aSI83DDUPQIAMvo36w9e9zN/E2A1AY/useROrfx4/zUgNJk1BNJtJ2pLaW8mHBOQbbflLJDedml6zwwFTtWr0vKvkzVU7goJQoMxDqFsy8HMGyxLfErNJaMNJKwLAy+0BUNyDWqtV8mklurOuyGIU1UeI86eKt7zEqzMeckmZ4fThzDZa7+hKh7O58JgKRue+SCgf3j+YcEtlYuKTatg1Srwqa3U3mSbQUVw8ELJbiUHneoiViYCDywSEKyCV56rxEQXHKSWfFMyUTLykpzWyMtB4LMiLD0Cr23f06xcmmTCDUKLOqs+stojJS0rNU0gsM0zIyMzPMUnRo0JPSWACO3WKW7cpNtXLQCTBuZippaYfjpJLjjZoWpSVYeUnOSjQeJaTLnG/Q1uuxGHJDe9PKQlS0Y45qjLSX2GAy7Lt+f5OfrVC+IxqwCqZQryi2jTmMd6fq05ZsQIa3Ut785hrUo9CUJLSpR6i9ZkR0yymaBadUbqM6rwrgva5JiGJUmM8hRpziM8xtOOKWUEj6TzSx5iIOhkPYVSsrOVumSS3t9ypNz20nrU26bqyUXqwUn/MIkdVU3Wc6SxpZpFBQ0+otSVrPFKfpNKzP6CAbaMlyccvOVv/Cv5dQDWhg25OpiKJTr1pL6CRU4VbWy+RlpzCSRI+zElmX0gIzJ9Wl23fWXCvsx3ZNHgPG+tprQTj6N8NREeoj8rOPmxIxfpU2l5SMiMqt12ilBZl059/epJEpbGaSs1xKzIv3SUlWBaMAEfuV61Vq5kkiu1tx15ceS5GjvOnipxlObhifPgZqT/ZF3ypcmV3e55nyVgODJFyX2r7uY+Ah9ZVXLabsKq8eDIrfUhKJB5qlGRmoiSaSSRnnZxpwMtR6QGR5Z6KcTIoSXLoqFZprrcdijw3GkIXJcWaSaz1JSSnDSgzURYFpSRniZCx16gWtR8n2T+2rynO02rsqjMU2XEIzdZmpSnFSFEkyIs49OcWbpL1GQVPLzacytVyyaDOrL9arM+pEbTZtIZRHiISZvLNCOc8Umaj/cwIi04+kQGW5UOVLJj7TL+FsakApuWO4HbWyYXHV4yjRIYimhlZfsuLMm0K+xSyP7Bx5EoaIGSO0Wm0kklU1l8yLnU4knDP7TUZgK/TadQqnlFup+w7jkUq4W97arTDcYnGHF6c1eatOGeWCizknhjjiR4njVsg9vt/8Aq/lCuCE69KpzS00xE15WcqS+WYp9eJFgeK0Eo8MC8YsNADfxl+R/zryie+FfesBp7hpJCjcwzMDzsdWAxmy6Xb0m0avPsO8J1Pst1992WwlhJFHwLF0mVrSS2kmWnnwxxTgYCoZGLSr6siM86BFjsKuGqKfNp6SqOrg/AkmlKySo0qVmqSR4H4qsS04D73QUu5GbXtGFPtyj02IzXIhRSiVRTxZyELJLeabCM1OHOWOGGowHoShvVJ+nIcrUKNCmmZ5zMaSchBFjo8c0Ix0eocd0+bNX9je+AwHDZXmbQfYI/wAtImQAVJHKy77kR89QC2iqXbk+tu7nVLuGHJmJWlKFNeHyG2jIjxL8GhZJx9eGIDt2pZ9GtRC26EzJYaUhLe9uTX30JSnUSUuLUSfsIhHVDJ7SZM2XKhSqvSVzFGuUimT3I7b6j1qUgjzSUfOoiIz9ICXtG1qNaFHRS7dgtwoaVGs0pxNS1HrUpR4mpWgtJnzEJoBl+5781Kx74kfcgac82h5pbTqErbWk0qSosSUR6yMgFEj5KLZaYjw3E1CRR4zhOsUp+a4uI2olGovwZngoiM/JViXqF9SRJSRJIiItBEXMApNxZLbRuOq8I1unSZcwl74haqhJIm1aNKEk4RI1F5JFqFrpNOj0mnMwYe/eDskZI359by8McdK1maj185mAzjdJ8lsr2ln4hqQCDuu1qXdDERFUbd32G8UmLIYdU06w4WpSFJ0l9Gox17csukUGpSKmymTLq8hBNuz5z6n31ILUklKPxU/0UkRAPm47KpddrEeruLmwquwycZE2DIUy7vRniaDMtBpx06S0HqwHcta16Ta0V9ijRjbOQ4b0h5xxTrr7h61uOKM1KP6T0cwCaGVRf1nZf1W/7KAGqgApVz5L7SumW5IuCnyZy1r3w0uVCSTaVYYYpQThJTo9BEJ+3Lep9uxXI9KTKSytWeZPy3pBkeGGg3FKMi0ai0AJYfD35Fz+qYDK9yzyF259Mn+ZdGqutoeaW26hK21kaVIUWJKI9ZGXOQCiPZKbckNNRJS6rIo7SyW3SXqg6qIkyPEi3sz0pI9STM0l6Baa1b9KrdH4KqUJp2B4ua0WKN7NPkmg04GgywLA0mRlzAK0WS+gPVGFMrD1VrSoK98itVSc5JbZV+8SVHgZ+tWIvIDKd0H+Y2N9aoXwujVgFQvjJtal8yo0i6aWqe7GQbbJ+FPNEhJniehCyL7deoRFByJ5PqBWIlVpNvkxPiOE6y6cyQvMUWo8FOGR/aQCy3JZ9Kr8+HUZKZEaqwyUmPPhvKZebSrWnOLykn+6ojL1D6ta0qVbTs6RBQ87PnrJcubKdN598yLBOcs+Yi0EksCLmIBPjLbO5esoHs8L5KAGpCp1uwqRVK25WGnahTKo82TUiTTJS4y5CC1E5m6FYaiMyxLmMBJ23a9GtuicE0eA0xBUajcQeKzdUrylLNWJrM+czxFal5KLekwl045Faboi/KpTdSeTFMscc0kY4pTj+yRkXqAXSl0+HSadHgU2M1Fhx0E20y0kkpQkuYiEbffmPcXu6R8pQCIyM8mFvez/APJQuYCOuCiU24qS/TK1EbmQXiLPacxw0HiRkZaSMj0kZaSFcYycUclxyqEus1WLGWlbMSo1Bx9hCk+SZoM8F4c2fnYALoADMMrfnlk696/7oGngIy5qHCuWgTqNVELXCmtG06SFZqsD9B8xj9tuiwrcoMCj0tCkQoTSWWiUrOPNL0nzmAhKxYNIqNclVht6pU6oTG0NS3qfMWwclCCwSS809JkWjEsDw5xNW3QKXbVIZplDhtw4TWJk2jEzMz1qUZ4mpR85mZmYCTGXZLeUnKX7ZH+5wBqIo9wZKbPuGcqZWqdKmPm6bxG5UpWahZniZoTvmCPoSREQCyUSgQKLTHIEApRRnFKUonpbz68TIiPBbilKLQWoj0cwqbmR6yXak3UHabNXUGyzUSlVaYbqS06CWbuJaz5+cwF4p0NmnQWIcXfN4ZQSEb66p1WBelSjNSj9ZmZhUv0dK/hL+4wEHkz5N7U90xPkoFkAAAAABUofKzVvckP58oBbQAUq58l9pXTLckXBT5M5a174aXKhJJtKsMMUoJwkp0egiE/blvU+3YrkelJlJZWrPMn5b0gyPDDQbilGRaNRaAEsADLtzhyat+2PfeQ091CXWltrxzVkaTwMyPA/WWkgFBcyPWS7Um6g7TZq6g2WaiUqrTDdSWnQSzdxLWfPzmLY9QYL1vlRXCleAEhLfizHku4JMjL8KSt8x0FpzsT5wFUh5ILKhT3Z0OmzWJjx4uvt1WWlbnP4yidxP7RfwGYboXzNp3vVj7ljTwAV+6bRpNzORH6i083Ohmaos2K8ph9gz15q0mR4HzkeJH6AHDQrLplJqhVRbs6pVRKDaRMqMlUhxpB60ox0II+fNIsecWYAGX0b9Yeve5m/ibAagKfdeTa17tkuv3DBkzTcNJqbVPkJaxSWBGTaXCQR4FzEAlbYtalWzHdYo7Ultl0kkaH5j0giJJGREknFqzS06iw/0FemZLaA/GkQ2JFYhUqTnb9TYlRdajKJXlETZHglJ6cSTgR46gFtoVHp9ApMemUaI1DgR05rTLRYJSWs/pMzxMzPSZniY+q7+hKh7O58JgKRue+SCgf3j+YcF+mxWJ0R+LMZbfjPoNt1pxOclaTLAyMucjIBSVZK7dfVFbqC6rUadFWTkenTp7r8Vsy8n8Go/GItRErEiIWe47eplx05MKrRt9ZQsnWlIWptbKy8laFpMlIUXMZGRgK9FyaUJNbhVapuVKszoOmGuqTFyCjniR4oSejO0F4xkZ6C06CF2AZTl2/P8nP1qhfEY1YBSr1yXWfe1TaqFz0lU6W00TKFnLfbJKCMzwJKFkWsz04YjqWxkcsO167FrFCoJRalGzjaeOW+5m5yTSfirWZHoUesgE5X7MpVZq7FXUcqDWWWjYRPgvmy7vZniaFYaFpx04KI8D1YDsWpatKtdiUmlsub9Ld3+VJfcU69Ic/eWtWJn9GouYiATgyXJxy85W/8K/l1ANaFTrdhUiqVtysNO1CmVR5smpEmmSlxlyEFqJzN0Kw1EZliXMYCTtu16NbdE4Jo8BpiCo1G4g8Vm6pXlKWasTWZ85niK1LyUW9JhLpxyK03RF+VSm6k8mKZY45pIxxSnH9kjIvUAulLp8Ok06PApsZqLDjoJtplpJJShJcxEILKlyZXd7nmfJWA4MkXJfavu5j4CE5cdDp1yUSXSK1GTKp8pGY60ozLEsSMtJaSMjIjIy1GRAIKl5PqLCq0SpSV1CqTYRGmGupS1yCjEZYfg0meaR6C8bDO9Yk7rtal3QxERVG3d9hvFJiyGHVNOsOFqUhSdJfRqMB17csukUGpSKmymTLq8hBNuz5z6n31ILUklKPxU/0UkRCyAMtyocqWTH2mX8LY1IBC3lbVOvC2ptCrSHFwJZJJwm15ivFUS0mR+o0kY79Ip0ekUmFTYKDREhsIjspM8TShCSSksefQRAKzU8ndFm1WoT2nqnAdqWb4eiBNcYRKwLAjWST14aMU4GfOLFQqPT6BSY9Mo0RqHAjpzW2WiwJJf7meszPSZ6TAd8Zfkf8AOvKJ74V96wGoCgpyT2wmOqCSKgVFU6b50gprhQzWas4/wZHpTnac0zzfUAvjLTbLSGmUJbaQkkoQgsCSRaCIi5iFPuvJnal2VA5tw0+RNfNSVESp8hKEGSSSRpQlwkpPAi1EWOvWAsdEpMWiU5uDT9/KO2Zmkn5Dj6ixPE/HcUpX+ugcd0+bNX9je+AwHDZXmbQfYI/y0iZABUkcrLvuRHz1ALaAAAAADL9z35qVj3xI+5A1AAAAABlu6T5LZXtLPxDUgAAAAAZVF/Wdl/Vb/soAaqAAAAPh78i5/VMBle5Z5C7c+mT/ADLo1cAAAABlO6D/ADGxvrVC+F0asAAAAADLbO5esoHs8L5KAGpAAAACDvvzHuL3dI+UoBEZGeTC3vZ/+ShcwAAAAGYZW/PLJ171/wB0DTwAAAAAZdkt5Scpftkf7nAGogAAADr1L9HSv4S/uMBB5M+Te1PdMT5KBZAAAAAAVKHys1b3JD+fKAW0AAAAAGXbnDk1b9se+8hqIAAAADMN0L5m073qx9yxp4AAAAAMvo36w9e9zN/E2A1AAAAAdKu/oSoezufCYCkbnvkgoH94/mHBogAAAADKcu35/k5+tUL4jGrAAAAAAyXJxy85W/8ACv5dQDWgAAABWMqXJld3ueZ8lYDgyRcl9q+7mPgIW4AAAABluVDlSyY+0y/hbGpAAAAAAy/I/wCdeUT3wr71gNQAAAAEZdPmzV/Y3vgMBw2V5m0H2CP8tImQAVauW/V37lRWaFV4UF3wQojiJUBUklJJZrIywdRhr9YDj4Mvbaeh9ROd6Dgy9tp6H1E53oA4Mvbaeh9ROd6Dgy9tp6H1E53oA4Mvbaeh9ROd6Dgy9tp6H1E53oBBWfY102rT5EOnXVSVtPyVylG7RFmecrDHDCSWjQQneDL22nofUTnegDgy9tp6H1E53oODL22nofUTnegDgy9tp6H1E53oODL22nofUTnegEFeti3Td9CXSandVJRGWtLhmzRFkrFJ4lpOSYneDL22nofUTnegDgy9tp6H1E53oODL22nofUTnegDgy9tp6H1E53oODL22nofUTnegDgy9tp6H1E53oQLdh3Qi+HLqK6qTwiuBweaeBF73veeS8cPCdeJFzgJ7gy9tp6H1E53oODL22nofUTnegDgy9tp6H1E53oODL22nofUTnegDgy9tp6H1E53ofiqXeykmk7noeBlh+g3O9AIGxLDuiyrWh0Ck3TSVQoprNs3qItS/HWpZ4mUkudRif4Mvbaeh9ROd6AODL22nofUTneg4Mvbaeh9ROd6AODL22nofUTneg4Mvbaeh9ROd6AQF32HdF1NUtuo3VSUlTp7VRZ3qiLLFxslEkjxkno8YxP8ABl7bT0PqJzvQBwZe209D6ic70HBl7bT0PqJzvQBwZe209D6ic70HBl7bT0PqJzvQBwZe209D6ic70IKm2LdNPuyr3CxdVJObU0NIeSqiLNBE2kklml4To0EQCd4Mvbaeh9ROd6Dgy9tp6H1E53oA4Mvbaeh9ROd6Dgy9tp6H1E53oA4Mvbaeh9ROd6HVqtv3lU6XMgSLoopMymVsLNNDcIySpJpPD8Z14GA61tWnd1vUKHSoV00dUeKjMQblDWajLEz0/jPrEnwZe209D6ic70AcGXttPQ+onO9BwZe209D6ic70AcGXttPQ+onO9BwZe209D6ic70Agrhsa6a7UqNOmXVSUu0p/whgm6IsiNWjyvxnSWghO8GXttPQ+onO9AHBl7bT0PqJzvQcGXttPQ+onO9AHBl7bT0PqJzvQcGXttPQ+onO9AHBl7bT0PqJzvQgqDYt00WtVuqRLqpKpFXcQ6+S6Is0kaSPDN/GdHlGAneDL22nofUTneg4Mvbaeh9ROd6AODL22nofUTneg4Mvbaeh9ROd6AODL22nofUTneh8PUi9XmVtrueiZq0mk8KG5qP8AvQCftumFRLdpdKJ3figxWopOZubn5iCTjhieGOGrExIgAAAAAqlZt6suXQ5WaFWYMFb0NqG63Kp6pOJIW4sjIydRh+UMsMD1APngy9tp6H1E53oODL22nofUTnegDgy9tp6H1E53oODL22nofUTnegDgy9tp6H1E53oODL22nofUTnegEFZdi3TaNFKl0y6qSuOTincXqIs1YqPTqkkJ3gy9tp6H1E53oA4Mvbaeh9ROd6Dgy9tp6H1E53oA4Mvbaeh9ROd6Dgy9tp6H1E53oBBXhY103XTWYVRuqkoaakIkJNqiLI85OOGuSejSYneDL22nofUTnegDgy9tp6H1E53oODL22nofUTnegDgy9tp6H1E53oODL22nofUTnegDgy9tp6H1E53oQUaxrpj3jLuRF1Uk58mMUVaToi8wkEZHoLwnHHxS5wE7wZe209D6ic70HBl7bT0PqJzvQBwZe209D6ic70HBl7bT0PqJzvQBwZe209D6ic70OKVRr0kxXmHLoohIdQaFYUNzHAyw/wDKARloWXddq27EotOumkKixs/MN2iLNR5y1LPEyklzqMTHBl7bT0PqJzvQBwZe209D6ic70HBl7bT0PqJzvQBwZe209D6ic70HBl7bT0PqJzvQCBuqw7ouZ2jOVC6qSk6VPaqLG90RZYuNniklfjOkhPcGXttPQ+onO9AHBl7bT0PqJzvQcGXttPQ+onO9AHBl7bT0PqJzvQcGXttPQ+onO9AHBl7bT0PqJzvQr9GsG56TddwXBGuqknOre8eEkuiLNBbyg0JzS8J0aD0gLBwZe209D6ic70HBl7bT0PqJzvQBwZe209D6ic70HBl7bT0PqJzvQBwZe209D6ic70OjXbavCtUSoUuXdFGKNOjuRnTRQ3CUSVpNJ4H4TrwMB829a130GhQKVDumjKjQmUsNm5Q3DUaUlgWJ+E6xIcGXttPQ+onO9AHBl7bT0PqJzvQcGXttPQ+onO9AHBl7bT0PqJzvQcGXttPQ+onO9AIKu2LdNartDq0q6qSmTSFuORyRRFkkzWSSPOLwnT5JCd4Mvbaeh9ROd6AODL22nofUTneg4Mvbaeh9ROd6AODL22nofUTneg4Mvbaeh9ROd6AODL22nofUTnehBW5Y100CoVmZDuqkqdqsk5T5OURZkSjx8n8Z0FpMBO8GXttPQ+onO9BwZe209D6ic70AcGXttPQ+onO9BwZe209D6ic70AcGXttPQ+onO9DhnUO85sKRFeueib0+2ppebQ3CPBRYHh+NesBZ6LBKmUeDAJzfCisNsZ+GGdmpJOOHNqHcAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAf/Z" style="width: 575px; height: 41px;" /></p>
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<p style="text-align: center;">Figure A.1: An array before and after a partition</p>
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<p>How a partition is executed and how a pivot element is selected are fascinating questions but are not of interest to us. What we would like you to do is the following: given an array, determine all the values that could be the pivot value assuming the array has been partitioned, or determine that the array has not been partitioned.</p>
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</article>
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<article class="section">
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<h2>입력</h2>
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<p>Input starts with a positive integer <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>n</mi></mrow></math></span> <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><msup><mn>10</mn><mn>6</mn></msup><mo stretchy="false">)</mo></mrow></math></span> denoting the size of the array. Following this are <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>n</mi></mrow></math></span> positive integers indicating the values in the array. All values are unique and <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo>≤</mo><msup><mn>10</mn><mn>6</mn></msup></mrow></math></span>.</p>
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</article>
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<article class="section">
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<h2>출력</h2>
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<p>Output <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>m</mi><mo>=</mo></mrow></math></span> the number of values in the array that could have served as pivot values to partition the array, followed by the pivot values in the order that they appear in the input. If <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>m</mi><mo>></mo><mn>100</mn></mrow></math></span> just output the first 100 of these pivot values. Note that a value of <span class="math-inline"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>m</mi><mo>=</mo><mn>0</mn></mrow></math></span> indicates that the array is not partitioned.</p>
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</article>
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<article class="section">
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<h2>예제 입력 1 복사</h2>
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<pre class="sampledata" id="sample-input-1">10 1 11 8 13 53 20 63 99 79 94
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</pre>
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</article>
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<article class="section">
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<h2>예제 출력 1 복사</h2>
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<pre class="sampledata" id="sample-output-1">3 1 13 63
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</pre>
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</article>
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</main>
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</body>
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</html>
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