{"trustable":true,"prependHtml":"\u003cscript\u003e window.katexOptions \u003d { disable: true }; \u003c/script\u003e\n\u003cscript type\u003d\"text/x-mathjax-config\"\u003e\n MathJax.Hub.Config({\n tex2jax: {\n inlineMath: [[\u0027$$$\u0027,\u0027$$$\u0027], [\u0027$\u0027,\u0027$\u0027]],\n displayMath: [[\u0027$$$$$$\u0027,\u0027$$$$$$\u0027], [\u0027$$\u0027,\u0027$$\u0027]]\n }\n });\n\u003c/script\u003e\n\u003cscript async src\u003d\"https://mathjax.codeforces.org/MathJax.js?config\u003dTeX-AMS-MML_HTMLorMML\" type\u003d\"text/javascript\"\u003e\u003c/script\u003e","sections":[{"title":"","value":{"format":"HTML","content":"\u003cdiv class\u003d\"panel_content\"\u003eTo be frank with you, this problem is a classic problem of tremendous magnitude which may increase the difficulty of this problem.\u003cbr\u003e\u003cbr\u003eWe define a type of operation concerning a positive integer $n$ $(n \u0026gt; 1)$ as to replace it with an integer $d$, one of factors of $n$ $(1 \\leq d \\leq n)$.\u003cbr\u003e\u003cbr\u003eYou are given a positive integer $n$ and then we will ask you to determine the expectation number of times to utilize this type of operation if we want to change $n$ into $1$ by operating again and again, assuming each possible $d$ in each operation has equal possibility to select.\u003cbr\u003e\u003cbr\u003eFor the sake of calculation, $n$ and all its distinct prime factors $p_1, p_2, \\cdots, p_m$ will be given, satisfying $n$ has $m$ distinct prime factors exactly.\u003c/div\u003e"}},{"title":"Input","value":{"format":"HTML","content":"The input contains multiple test cases.\u003cbr\u003e\u003cbr\u003eFor each test case:\u003cbr\u003e\u003cbr\u003eThe first line contains two positive integers $n$ and $m$ which indicates $m$ is the number of distinct prime factors of $n$, satisfying $2 \\leq n \\leq 10^{24}$.\u003cbr\u003e\u003cbr\u003eThe second lines contains $m$ distinct prime numbers $p_1, p_2, \\cdots, p_m$, satisfying $2 \\leq p_i \\leq 10^6$.\u003cbr\u003e\u003cbr\u003eAbout $2 \\cdot 10^5$ test cases in total.\u003cbr\u003e\u003cbr\u003eWarm Tips for C/C++: \u003cb\u003e__int128_t\u003c/b\u003e is available here but standard solutions of this problem do not use this compiler-dependent data type."}},{"title":"Output","value":{"format":"HTML","content":"For each test case, output \"\u003cb\u003eCase #$x$: $y$\u003c/b\u003e\" in one line (without quotes), where $x$ indicates the case number starting from $1$ and $y$ denotes the expectation number of times to utilize this type of operation of corresponding case. Your answer will be considered correct if its absolute or relative error won\u0027t exceed $10^{-9}$."}},{"title":"Sample","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e2 1\r\n2\r\n4 1\r\n2\r\n6 2\r\n2 3\r\n8 1\r\n2\r\n10 2\r\n2 5\r\n12 2\r\n2 3\r\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003eCase #1: 2.0000000000\r\nCase #2: 2.5000000000\r\nCase #3: 2.6666666667\r\nCase #4: 2.8333333333\r\nCase #5: 2.6666666667\r\nCase #6: 3.0333333333\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}}]}