CF1228B.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 $1000000007\,(10^{9} + 7)$.

假设有一个 $h \times w$ 的网格，每个格子为空或满。我们给出如下定义：

• $r_{i}$ 表示第 $i$ 行（$1 \le i \le h$）中与左边界相连的连续满格子的个数。特别地，若第 $i$ 行最左侧的格子为空，则 $r_i = 0$。
• $c_{j}$ 表示第 $j$ 列（$1 \le j \le w$）中与上边界相连的连续满格子的个数。特别地，若第 $j$ 列最上方的格子为空，则 $c_j = 0$。

换言之，第 $i$ 行恰好以 $r_i$ 个满格子开头；第 $j$ 列恰好以 $c_j$ 个满格子开头。

这是一个 $3 \times 4$ 网格对应的 $r$ 和 $c$ 值示例。黑色格子为满格子，白色格子为空格子。

已知 $r$ 和 $c$ 的值，初始时所有格子均为空。求满足给定 $r$ 和 $c$ 值的填法总数。由于答案可能非常大，请对 $1000000007\,(10^{9} + 7)$ 取模输出结果，即输出答案除以 $1000000007\,(10^{9} + 7)$ 后的余数。

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 $c$.

第一行包含两个整数 $h$ 和 $w$（$1 \le h, w \le 10^{3}$）——网格的高度和宽度。

第二行包含 $h$ 个整数 $r_{1}, r_{2}, \ldots, r_{h}$（$0 \le r_{i} \le w$）——$r$ 的值。

第三行包含 $w$ 个整数 $c_{1}, c_{2}, \ldots, c_{w}$（$0 \le c_{j} \le h$）——$c$ 的值。

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

对 $1000000007\,(10^{9} + 7)$ 取模后输出答案。

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 $(10^9 + 7)$.

在第一个例子中，这是另一种可能的情况。

在第二个例子中，无法构造出满足给定 $r$、$c$ 值的网格。

在第三个例子中，请确保输出答案对 $(10^9 + 7)$ 取模的结果。