Problem Statement

Given is an integer $N$. How many permutations $(P_0,P_1,\cdots,P_{2N-1})$ of $(0,1,\cdots,2N-1)$ satisfy the following condition?

• For each $i$ $(0 \leq i \leq 2N-1)$, $N^2 \leq i^2+P_i^2 \leq (2N)^2$ holds.

Since the number can be enormous, compute it modulo $M$.

Constraints

• $1 \leq N \leq 250$
• $2 \leq M \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M

Output

Print the number of permutations that satisfy the condition, modulo $M$.