Problem Statement

Find the number, modulo $998244353$, of square matrices of size $N$ whose elements are non-negative integers, that satisfy both of the following two conditions:

• for all $i = 1, 2, \\ldots, N$, the sum of the elements in the $i$-th row is $R_i$;
• for all $i = 1, 2, \\ldots, N$, the sum of the elements in the $i$-th column is $C_i$.

Note that $R_i$ and $C_i$ given in the input are integers between $0$ and $2$ (see Constraints).

Constraints

• $1 \\leq N \\leq 5000$
• $0 \\leq R_i \\leq 2$
• $0 \\leq C_i \\leq 2$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $R_1$ $R_2$ $\\ldots$ $R_N$ $C_1$ $C_2$ $\\ldots$ $C_N$

Output

Print the answer.

Sample Input 1

3
1 1 1
0 1 2

Sample Output 1

3

The following $3$ matrices satisfy the conditions:

0 1 0
0 0 1
0 0 1
``````plain
0 0 1
0 1 0
0 0 1
``````plain
0 0 1
0 0 1
0 1 0

Sample Input 2

3
1 1 1
2 2 2

Sample Output 2

0

Sample Input 3

18
2 0 1 2 0 1 1 2 1 1 2 0 1 2 2 1 0 0
1 1 0 1 1 1 1 1 1 1 1 1 2 1 1 0 2 2

Sample Output 3

968235177

Be sure to print the count modulo $998244353$.