200843: [AtCoder]ARC084 F - XorShift

Problem Statement

There are $N$ non-negative integers written on a blackboard. The $i$-th integer is $A_i$.

Takahashi can perform the following two kinds of operations any number of times in any order:

• Select one integer written on the board (let this integer be $X$). Write $2X$ on the board, without erasing the selected integer.
• Select two integers, possibly the same, written on the board (let these integers be $X$ and $Y$). Write $X$ XOR $Y$ (XOR stands for bitwise xor) on the blackboard, without erasing the selected integers.

How many different integers not exceeding $X$ can be written on the blackboard? We will also count the integers that are initially written on the board. Since the answer can be extremely large, find the count modulo $998244353$.

Constraints

• $1 \leq N \leq 6$
• $1 \leq X < 2^{4000}$
• $1 \leq A_i < 2^{4000}(1\leq i\leq N)$
• All input values are integers.
• $X$ and $A_i(1\leq i\leq N)$ are given in binary notation, with the most significant digit in each of them being $1$.

Input

Input is given from Standard Input in the following format:

$N$ $X$
$A_1$
$:$
$A_N$

Output

Print the number of different integers not exceeding $X$ that can be written on the blackboard.

Sample Input 1

3 111
1111
10111
10010

Sample Output 1

4

Initially, $15$, $23$ and $18$ are written on the blackboard. Among the integers not exceeding $7$, four integers, $0$, $3$, $5$ and $6$, can be written. For example, $6$ can be written as follows:

• Double $15$ to write $30$.
• Take XOR of $30$ and $18$ to write $12$.
• Double $12$ to write $24$.
• Take XOR of $30$ and $24$ to write $6$.

Sample Input 2

4 100100
1011
1110
110101
1010110

Sample Output 2

37

Sample Input 3

4 111001100101001
10111110
1001000110
100000101
11110000011

Sample Output 3

1843

Sample Input 4

1 111111111111111111111111111111111111111111111111111111111111111
1

Sample Output 4

466025955

Be sure to find the count modulo $998244353$.