Score : 1800 points

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.

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