Problem Statement

We have a checkerboard with $H$ rows and $W$ columns, where each square has a 0 or 1 written on it. The current state of checkerboard is represented by $H$ strings $S_1,S_2,\cdots,S_H$: the $j$-th character of $S_i$ represents the digit in the square at the $i$-th row from the top and $j$-th column from the left.

Snuke will repeat the following operation.

• Choose one square uniformly at random. Then, flip the value written in that square. (In other words, change 0 to 1 and vice versa.)

By the way, he loves an integer sequence $A=(A_1,A_2,\cdots,A_H)$, so he will terminate the process at the moment when the following condition is satisfied.

• For every $i$ ($1 \leq i \leq H$), the $i$-th row from the top contains exactly $A_i$ 1s.

Particularly, he may do zero operations.

Find the expected value of the number of operations Snuke does, modulo $998244353$.

Definition of the expected value modulo $998244353$ It can be proved that the sought expected value is always a rational number. Additionally, under the Constraints of this problem, when that value is represented as an irreducible fraction $\frac{P}{Q}$, it can also be proved that $Q \not \equiv 0 \pmod{998244353}$. Thus, there uniquely exists an integer $R$ such that $R \times Q \equiv P \pmod{998244353}, 0 \leq R < 998244353$. Find this $R$.

Constraints

• $1 \leq H,W \leq 50$
• $S_i$ is a string of length $W$ consisting of 0, 1.
• $0 \leq A_i \leq W$

Input

Input is given from Standard Input in the following format:

H W
S_1
S_2
\vdots
S_H
A_1 A_2 \cdots A_H

Output

Print the answer.