A12913. Filling the Grid

Suppose there is a $h \times w$ grid consisting of empty or full cells. Let's make some definitions:

- $r_{i}$ is the number of consecutive full cells connected to the left side in the $i$ -th row ( $1 \le i \le h$ ). In particular, $r_i=0$ if the leftmost cell of the $i$ -th row is empty.
- $c_{j}$ is the number of consecutive full cells connected to the top end in the $j$ -th column ( $1 \le j \le w$ ). In particular, $c_j=0$ if the topmost cell of the $j$ -th column is empty.

In other words, the $i$ -th row starts exactly with $r_i$ full cells. Similarly, the $j$ -th column starts exactly with $c_j$ full cells.

These are the $r$ and $c$ values of some $3 \times 4$ grid. Black cells are full and white cells are empty.You have values of $r$ and $c$ . Initially, all cells are empty. Find the number of ways to fill grid cells to satisfy values of $r$ and $c$ . Since the answer can be very large, find the answer modulo $1000000007\,(10^{9} + 7)$ . In other words, find the remainder after division of the answer by $1000000007\,(10^{9} + 7)$ .

The first line contains two integers $h$ and $w$ ( $1 \le h, w \le 10^{3}$ ) — the height and width of the grid.

The second line contains $h$ integers $r_{1}, r_{2}, \ldots, r_{h}$ ( $0 \le r_{i} \le w$ ) — the values of $r$ .

The third line contains $w$ integers $c_{1}, c_{2}, \ldots, c_{w}$ ( $0 \le c_{j} \le h$ ) — the values of $c$ .

Print the answer modulo $1000000007\,(10^{9} + 7)$ .

In the first example, this is the other possible case.

In the second example, it's impossible to make a grid to satisfy such $r$ , $c$ values.

In the third example, make sure to print answer modulo $(10^9 + 7)$ .