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G - Row Column Sums 2

Score : $600$ points

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.

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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.

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3
1 1 1
0 1 2

3

The following $3$ matrices satisfy the conditions:

``` 0 1 0 0 0 1 0 0 1 ``` ``` 0 0 1 0 1 0 0 0 1 ``` ``` 0 0 1 0 0 1 0 1 0 ```

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3
1 1 1
2 2 2

0

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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

968235177

Be sure to print the count modulo $998244353$.