Filling the Grid
Description
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 $1000000007\,(10^{9} + 7)$.
Input
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 $c$.
Output
Print the answer modulo $1000000007\,(10^{9} + 7)$.
Samples
3 4
0 3 1
0 2 3 0
2
1 1
0
1
0
19 16
16 16 16 16 15 15 0 5 0 4 9 9 1 4 4 0 8 16 12
6 12 19 15 8 6 19 19 14 6 9 16 10 11 15 4
797922655
Note
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 $(10^9 + 7)$.