Score : $1200$ points

Problem Statement

Given is a positive integer $N$. Find the number of permutations $(P_1,P_2,\cdots,P_{3N})$ of $(1,2,\cdots,3N)$ that can be generated through the procedure below. This number can be enormous, so print it modulo a prime number $M$.

• Make $N$ sequences $A_1,A_2,\cdots,A_N$ of length $3$ each, using each of the integers $1$ through $3N$ exactly once.
• Let $P$ be an empty sequence, and do the following operation $3N$ times.- Among the elements that are at the beginning of one of the sequences $A_i$ that is non-empty, let the smallest be $x$.
  • Remove $x$ from the sequence, and add $x$ at the end of $P$.
• Among the elements that are at the beginning of one of the sequences $A_i$ that is non-empty, let the smallest be $x$.
• Remove $x$ from the sequence, and add $x$ at the end of $P$.

Constraints

• $1 \leq N \leq 2000$
• $10^8 \leq M \leq 10^9+7$
• $M$ is a prime number.
• 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 modulo $M$.

1 998244353

6

All permutations of length $3$ count.

2 998244353

261

314 1000000007

182908545