{"problem":{"name":"Only Once","description":{"content":"For a sequence of length $N$, $A = (A_1,A_2,\\dots,A_N)$, consisting of integers between $1$ and $N$, inclusive, and an integer $i\\ (1\\leq i \\leq N)$, let us define a sequence of length $10^{100}$, $B_","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"abc284_g"},"statements":[{"statement_type":"Markdown","content":"For a sequence of length $N$, $A = (A_1,A_2,\\dots,A_N)$, consisting of integers between $1$ and $N$, inclusive, and an integer $i\\ (1\\leq i \\leq N)$, let us define a sequence of length $10^{100}$, $B_i=(B_{i,1},B_{i,2},\\dots,B_{i,10^{100}})$, as follows.\n\n*   $B_{i,1}=i$.\n*   $B_{i,j+1}=A_{B_{i,j}}\\ (1\\leq j<10^{100})$.\n\nAdditionally, let us define $S_i$ as the number of distinct integers that occur exactly once in the sequence $B_i$. More formally, $S_i$ is the number of values $k$ such that exactly one index $j\\ (1\\leq j\\leq 10^{100})$ satisfies $B_{i,j}=k$.\nYou are given an integer $N$. There are $N^N$ sequences that can be $A$. Find the sum of $\\displaystyle \\sum_{i=1}^{N} S_i$ over all of them, modulo $M$.\n\n## Constraints\n\n*   $1\\leq N \\leq 2\\times 10^5$\n*   $10^8\\leq M \\leq 10^9$\n*   $N$ and $M$ are integers.\n\n## Input\n\nThe input is given from Standard Input in the following format:\n\n$N$ $M$\n\n[samples]","is_translate":false,"language":"English"},{"statement_type":"Markdown","content":"对于一个长度为 $N$ 的序列 $A=(A_1,A_2,\\dots,A_N)$，其由 $1$ 到 $N$（含）之间的整数构成，以及一个整数 $i\\ (1\\leq i \\leq N)$，定义一个长度为 $10^{100}$ 的序列 $B_i=(B_{i,1},B_{i,2},\\dots,B_{i,10^{100}})$ 如下。\n\n*   $B_{i,1}=i$.\n*   $B_{i,j+1}=A_{B_{i,j}}\\ (1\\leq j<10^{100})$.\n\n此外，定义 $S_i$ 为在序列 $B_i$ 中恰好出现一次的不同整数的个数。更正式地，$S_i$ 是满足恰好存在一个下标 $j\\ (1\\leq j\\leq 10^{100})$ 使得 $B_{i,j}=k$ 的值 $k$ 的个数。\n给定一个整数 $N$。$A$ 可能的序列共有 $N^N$ 个。求对所有这些序列，$\\displaystyle \\sum_{i=1}^{N} S_i$ 的总和，并对 $M$ 取模。\n\n## Constraints\n\n*   $1\\leq N \\leq 2\\times 10^5$\n*   $10^8\\leq M \\leq 10^9$\n*   $N$ 和 $M$ 为整数。\n\n## Input\n\n输入从标准输入给出，格式如下：\n\n$N$ $M$\n\n[samples]","is_translate":true,"language":"Chinese"}],"meta":{"iden":"abc284_g","tags":[],"sample_group":[["4 100000000","624\n\nAs an example, let us consider the case $A=(2,3,3,4)$.\n\n*   For $i=1$: we have $B_1=(1,2,3,3,3,\\dots)$, where two integers, $1$ and $2$, occur exactly once, so $S_1=2$.\n*   For $i=2$: we have $B_2=(2,3,3,3,\\dots)$, where one integer, $2$, occurs exactly once, so $S_2=1$.\n*   For $i=3$: we have $B_3=(3,3,3,\\dots)$, where no integers occur exactly once, so $S_3=0$.\n*   For $i=4$: we have $B_4=(4,4,4,\\dots)$, where no integers occur exactly once, so $S_4=0$.\n\nThus, we have $\\displaystyle \\sum_{i=1}^{N} S_i=2+1+0+0=3$.\nIf we similarly compute $\\displaystyle \\sum_{i=1}^{N} S_i$ for the other $255$ sequences, the sum of this value over all $256$ sequences is $624$."],["7 1000000000","5817084"],["2023 998244353","737481389\n\nPrint the sum modulo $M$."],["100000 353442899","271798911"]],"created_at":"2026-03-03 11:01:13"}}