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 $i (1 \leq i \leq N)$.

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.

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.

Sample Input 1

4
1 2 3 4
2 1 4 3

Sample Output 1

4

There are four valid permutations: (3, 4, 1, 2), (3, 4, 2, 1), (4, 3, 1, 2), and (4, 3, 2, 1).

Sample Input 2

3
1 2 3
2 1 3

Sample Output 2

0

The answer may be 0.

Sample Input 3

20
2 3 15 19 10 7 5 6 14 13 20 4 18 9 17 8 12 11 16 1
8 12 4 13 19 3 10 16 11 9 1 2 17 6 5 18 7 14 20 15

Sample Output 3

803776944

Be sure to print the count modulo (10^9 + 7).

见题面 Input 部分

见题面 Output 部分