문제

Farmer John has come up with a new morning exercise routine for the cows (again)!

As before, Farmer John's $N$ cows ($1\le N\le 7500$) are standing in a line. The $i$-th cow from the left has label $i$ for each $1\le i\le N$. He tells them to repeat the following step until the cows are in the same order as when they started.

Given a permutation $A$ of length $N$, the cows change their order such that the $i$-th cow from the left before the change is $A_i$-th from the left after the change.

For example, if $A=(1,2,3,4,5)$ then the cows perform one step and immediately return to the same order. If $A=(2,3,1,5,4)$, then the cows perform six steps before returning to the original order. The order of the cows from left to right after each step is as follows:

0 steps: $(1,2,3,4,5)$1 step: $(3,1,2,5,4)$2 steps: $(2,3,1,4,5)$3 steps: $(1,2,3,5,4)$4 steps: $(3,1,2,4,5)$5 steps: $(2,3,1,5,4)$6 steps: $(1,2,3,4,5)$

Compute the product of the numbers of steps needed over all $N!$ possible permutations $A$ of length $N$.

As this number may be very large, output the answer modulo $M$ ($10^8\le M\le 10^9+7$, $M$ is prime).

Contestants using C++ may find the following code from KACTL helpful. Known as the Barrett reduction, it allows you to compute $a \% b$ several times faster than usual, where $b>1$ is constant but not known at compile time. (we are not aware of such an optimization for Java, unfortunately).

#include <bits/stdc++.h> using namespace std;

typedef unsigned long long ull; typedef __uint128_t L; struct FastMod { ull b, m; FastMod(ull b) : b(b), m(ull((L(1) << 64) / b)) {} ull reduce(ull a) { ull q = (ull)((L(m) * a) >> 64); ull r = a - q * b; // can be proven that 0 <= r < 2*b return r >= b ? r - b : r; } }; FastMod F(2);

int main() { int M = 1000000007; F = FastMod(M); ull x = 10ULL*M+3; cout << x << " " << F.reduce(x) << "\n"; // 10000000073 3 }

입력

The first line contains $N$ and $M$.

출력

A single integer.

예제 입력 1

5 1000000007

예제 출력 1

369329541

점수

Test case 2 satisfies $N=8$.Test cases 3-5 satisfy $N\le 50$.Test cases 6-8 satisfy $N\le 500$.Test cases 9-12 satisfy $N\le 3000$.Test cases 13-16 satisfy no additional constraints.