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G - Only Once

Score : $600$ points

Problem Statement

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.

• $B_{i,1}=i$.
• $B_{i,j+1}=A_{B_{i,j}}\ (1\leq j<10^{100})$.

Additionally, 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 $B_{i,j}=k$.

You 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 $M$.

Constraints

• $1\leq N \leq 2\times 10^5$
• $10^8\leq M \leq 10^9$
• $N$ and $M$ are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$

Output

Print the answer as an integer.

4 100000000

624

As an example, let us consider the case $A=(2,3,3,4)$.

• For $i=1$: we have $B_1=(1,2,3,3,3,\dots)$, where two integers, $1$ and $2$, occur exactly once, so $S_1=2$.
• For $i=2$: we have $B_2=(2,3,3,3,\dots)$, where one integer, $2$, occurs exactly once, so $S_2=1$.
• For $i=3$: we have $B_3=(3,3,3,\dots)$, where no integers occur exactly once, so $S_3=0$.
• For $i=4$: we have $B_4=(4,4,4,\dots)$, where no integers occur exactly once, so $S_4=0$.

Thus, we have $\displaystyle \sum_{i=1}^{N} S_i=2+1+0+0=3$.

If 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

Print the sum modulo $M$.

100000 353442899

271798911