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.

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.

Constraints

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

Sample Input 1

2 998244353

Sample Output 1

4

Four permutations satisfy the condition:

• (2,3,0,1)
• (2,3,1,0)
• (3,2,0,1)
• (3,2,1,0)

Sample Input 2

10 998244353

Sample Output 2

53999264

Sample Input 3

200 998244353

Sample Output 3

112633322

见题面 Input 部分

见题面 Output 部分