Entanglement

Consider an array $A$ of length $N$ and an array $B$ of length $M$. An entanglement of these two arrays is a matrix $C$ of size $N \times M$ such that for all $0 \le i \le N - 1$ and $0 \le j \le M - 1$, at least one of the following conditions holds: $C\[i]\[j] = A\[i]$ or $C\[i]\[j] = B\[j]$.

You are given a matrix $C$ of size $N \times M$ and a number $K$. Your task is to count the number of pairs of arrays $(A, B)$ such that:

• $A$ has length $N$.
• $B$ has length $M$.
• $A$ and $B$ consist of values from the set $\\{1, 2, \ldots, K\\}$.
• $C$ is an entanglement of $A$ and $B$.

Print the number of such pairs modulo $10^9 + 7$.

The first line of input contains three integers $N$, $M$ and $K$ ($1 \le N, M \le 300$, $1 \le K \le N \times M$).

Each of the following $N$ lines contains $M$ integers separated by spaces, the $j$-th number on the $i$-th of these lines is $C\[i - 1]\[j - 1]$.

Print a single line containing a single integer: the number of pairs of arrays $(A, B)$ modulo $10^9 + 7$.