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.