Problem Statement

Given are permutations of $(1, \dots, N)$: $p = (p_1, \dots, p_N)$ and $q = (q_1, \dots, q_N)$.

Find the number, modulo $(10^9 + 7)$, of permutations $r = (r_1, \dots, r_N)$ of $(1, \dots, N)$ such that $r_i \neq p_i$ and $r_i \neq q_i$ for every $i$ $(1 \leq i \leq N)$.

Constraints

• $1 \leq N \leq 3000$
• $1 \leq p_i, q_i \leq N$
• $p_i \neq p_j \, (i \neq j)$
• $q_i \neq q_j \, (i \neq j)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
p_1 \ldots p_N
q_1 \ldots q_N

Output

Print the answer.