Xomu - Last Dance⠀

You are given an integer $n$.

For each $(m, k)$ such that $3 \leq m \leq n+1$ and $0 \leq k \leq n-1$, count the permutations of $[1, 2, ..., n]$ such that $p_i + p_{i+1} \geq m$ for exactly $k$ indices $i$, modulo $998\,244\,353$.

Input

The input consists of a single line, which contains two integers $n$, $x$ ($2 \leq n \leq 4000$, $1 \leq x \lt 1\,000\,000\,007$).

Output

Let $a_{m,k}$ be the answer for the pair $(m, k)$, modulo $998\,244\,353$.

Let $$\large S = \sum_{m=3}^{n+1} \sum_{k=0}^{n-1} a_{m,k}x^{mn+k}\phantom{0}.$$

Output a single line with an integer: $S$ modulo $1\,000\,000\,007$.

Note that using two different modulos is intentional. We want you to calculate all the $a_{m,k}$ modulo $998\,244\,353$, then treat them like integers in the range $[0, 998\,244\,352]$, and hash them modulo $1\,000\,000\,007$.

Note

In the first test case, the answers for all $(m, k)$ are shown in the following table:

$k = 0$$k = 1$$k = 2$$m = 3$$0$$0$$6$$m = 4$$0$$4$$2$

• The answer for $(m, k) = (3, 2)$ is $6$, because for every permutation of length $3$, $a_i + a_{i+1} \geq 3$ exactly $2$ times.
• The answer for $(m, k) = (4, 2)$ is $2$. In fact, there are $2$ permutations of length $3$ such that $a_i + a_{i+1} \geq 4$ exactly $2$ times: $[1, 3, 2]$, $[2, 3, 1]$.

Therefore, the value to print is $2^9 \cdot 0 + 2^{10} \cdot 0 + 2^{11} \cdot 6 + 2^{12} \cdot 0 + 2^{13} \cdot 4 + 2^{14} \cdot 2 \equiv 77\,824 \phantom{0} (\text{mod} \phantom{0} 1\,000\,000\,007)$.

In the second test case, the answers for all $(m, k)$ are shown in the following table:

$k = 0$$k = 1$$k = 2$$k = 3$$m = 3$$0$$0$$0$$24$$m = 4$$0$$0$$12$$12$$m = 5$$0$$4$$16$$4$

• The answer for $(m, k) = (5, 1)$ is $4$. In fact, there are $4$ permutations of length $4$ such that $a_i + a_{i+1} \geq 5$ exactly $1$ time: $[2, 1, 3, 4]$, $[3, 1, 2, 4]$, $[4, 2, 1, 3]$, $[4, 3, 1, 2]$.

In the third test case, the answers for all $(m, k)$ are shown in the following table:

$k = 0$$k = 1$$k = 2$$k = 3$$k = 4$$k = 5$$k = 6$$k = 7$$m = 3$$0$$0$$0$$0$$0$$0$$0$$40320$$m = 4$$0$$0$$0$$0$$0$$0$$10080$$30240$$m = 5$$0$$0$$0$$0$$0$$1440$$17280$$21600$$m = 6$$0$$0$$0$$0$$480$$8640$$21600$$9600$$m = 7$$0$$0$$0$$96$$3456$$16416$$16896$$3456$$m = 8$$0$$0$$48$$2160$$12960$$18240$$6480$$432$$m = 9$$0$$16$$1152$$9648$$18688$$9648$$1152$$16$