XorShift

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.

Examples

Input

3 111 1111 10111 10010

Output

4

Input

4 100100 1011 1110 110101 1010110

Output

37

Input

4 111001100101001 10111110 1001000110 100000101 11110000011

Output

1843

Input

1 111111111111111111111111111111111111111111111111111111111111111 1

Output

466025955

inputFormat

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

outputFormat

Output

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

Examples

Input

3 111 1111 10111 10010

Output

4

Input

4 100100 1011 1110 110101 1010110

Output

37

Input

4 111001100101001 10111110 1001000110 100000101 11110000011

Output

1843

Input

1 111111111111111111111111111111111111111111111111111111111111111 1

Output

466025955

样例

1 111111111111111111111111111111111111111111111111111111111111111
1

466025955