Score : $1000$ points

Problem Statement

You are given an integer sequence of length $N$: $A_1,A_2,...,A_N$. Let us perform $Q$ operations in order. The $i$-th operation is described by two integers $X_i$ and $Y_i$. In this operation, we will choose one of the following two actions and perform it:

• Swap the values of $A_{X_i}$ and $A_{Y_i}$
• Do nothing

There are $2^Q$ ways to perform these operations. Find the sum of the inversion numbers of the final sequences for all of these ways to perform operations, modulo $10^9+7$.

Here, the inversion number of a sequence $P_1,P_2,...,P_M$ is the number of pairs of integers $(i,j)$ such that $1\leq i < j\leq M$ and $P_i > P_j$.

Constraints

• $1 \leq N \leq 3000$
• $0 \leq Q \leq 3000$
• $0 \leq A_i \leq 10^9(1\leq i\leq N)$
• $1 \leq X_i,Y_i \leq N(1\leq i\leq Q)$
• $X_i\neq Y_i(1\leq i\leq Q)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $Q$

$A_1$

$:$

$A_N$

$X_1$ $Y_1$

$:$

$X_Q$ $Y_Q$

Output

Print the sum of the inversion numbers of the final sequences, modulo $10^9+7$.

3 2
1
2
3
1 2
1 3

6

There are four ways to perform operations, as follows:

• Do nothing, both in the first and second operations. The final sequence would be $1,2,3$, with the inversion number of $0$.
• Do nothing in the first operation, then perform the swap in the second. The final sequence would be $3,2,1$, with the inversion number of $3$.
• Perform the swap in the first operation, then do nothing in the second. The final sequence would be $2,1,3$, with the inversion number of $1$.
• Perform the swap, both in the first and second operations. The final sequence would be $3,1,2$, with the inversion number of $2$.

The sum of these inversion numbers, $0+3+1+2=6$, should be printed.

5 3
3
2
3
1
4
1 5
2 3
4 2

36

9 5
3
1
4
1
5
9
2
6
5
3 5
8 9
7 9
3 2
3 8

425