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Ex - E or m

Score : $600$ points

Problem Statement

We have a grid $A$ with $N$ rows and $M$ columns. Initially, $0$ is written on every square.
Let us perform the following operation.

• For each integer $i$ such that $1 \le i \le N$, in the $i$-th row, turn the digits in zero or more leftmost squares into $1$.
• For each integer $j$ such that $1 \le j \le M$, in the $j$-th column, turn the digits in zero or more topmost squares into $1$.

Let $S$ be the set of grids that can be obtained in this way.

You are given a grid $X$ with $N$ rows and $M$ columns consisting of 0, 1, and ?.
There are $2^q$ grids that can be obtained by replacing each ? with 0 or 1, where $q$ is the number of ? in $X$. How many of them are in $S$?
This count can be enormous, so find it modulo $998244353$.

Constraints

• $N$ and $M$ are integers.
• $1 \le N,M \le 18$
• $X$ is a grid with $N$ rows and $M$ columns consisting of 0, 1, and ?.

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Input

The input is given from Standard Input in the following format:

$N$ $M$

$X_{1,1} X_{1,2} \dots X_{1,M}$

$X_{2,1} X_{2,2} \dots X_{2,M}$

$\vdots$

$X_{N,1} X_{N,2} \dots X_{N,M}$

Output

Print an integer representing the answer.

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2 3
0?1
?1?

6

The following six grids are in $S$.

``` 011 011 001 010 011 110

001 011 011 111 110 111

<hr />




```input2
5 3
101
010
101
010
101

0

$X$ may have no ?, and the answer may be $0$.

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18 18
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????
??????????????????

462237431

Be sure to find the count modulo $998244353$.