For two permutations $A$ and $B$ of length $n$, Randias can generate a permutation $C$ of length $n$ as $C = A \circ B$ in this way: for each $1\le i \le n$, $C[i] = A[B[i]]$.

Now he is given $m$ permutations $A_{1},A_{2}, \dots, A_{m}$, each of them is of length $n$. He wants to choose a non-empty set of indices $i_{1},i_{2}, \dots, i_{k}$ ($1\le k \le m$,$1\le i_{1} < i_{2} \dots < i_{k} \le m$), and calculate $C = (((A_{i_{1}} \circ A_{i_{2}}) \circ A_{i_{3}}) \circ A_{i_{4}}) \dots \circ A_{i_{k}}$. Randias wants to know, how many possible permutations $C$ he can generate? Output the answer modulo $10^9 + 7$.

A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array), and $[1,3,4]$ is also not a permutation ($n=3$ but there is $4$ in the array)