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brute force

dp

fft

math

number theory

*3500

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H2. Cyclic Hamming (Hard Version)

time limit per test

2 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputThis is the hard version of the problem. The only difference between the two versions is the constraint on $$$k$$$. You can make hacks only if all versions of the problem are solved.

In this statement, all strings are $$$0$$$-indexed.

For two strings $$$a$$$, $$$b$$$ of the same length $$$p$$$, we define the following definitions:

- The hamming distance between $$$a$$$ and $$$b$$$, denoted as $$$h(a, b)$$$, is defined as the number of positions $$$i$$$ such that $$$0 \le i < p$$$ and $$$a_i \ne b_i$$$.
- $$$b$$$ is a cyclic shift of $$$a$$$ if there exists some $$$0 \leq k < p$$$ such that $$$b_{(i+k) \bmod p} = a_i$$$ for all $$$0 \le i < p$$$. Here $$$x \bmod y$$$ denotes the remainder from dividing $$$x$$$ by $$$y$$$.

You are given two binary strings $$$s$$$ and $$$t$$$ of length $$$2^{k+1}$$$ each. Both strings may contain missing characters (denoted by the character '?'). Your task is to count the number of ways to replace the missing characters in both strings with the characters '0' or '1' such that:

- Each string $$$s$$$ and $$$t$$$ contains exactly $$$2^k$$$ occurrences of each character '0' and '1'
- $$$h(s, c) \ge 2^k$$$ for all strings $$$c$$$ that is a cyclic shift of $$$t$$$.

As the result can be very large, you should print the value modulo $$$998\,244\,353$$$.

Input

The first line of the input contains a single integer $$$k$$$ ($$$1 \le k \le 12$$$).

The second line of the input contains string $$$s$$$ of size $$$2^{k+1}$$$, consisting of the characters '0', '1' and '?'.

The third line of the input contains string $$$t$$$ of size $$$2^{k+1}$$$, consisting of the characters '0', '1' and '?'.

It is guaranteed that both strings $$$s$$$ and $$$t$$$ contains no more than $$$2^k$$$ character '0' or '1'.

Output

Print a single integer — the answer to the problem modulo $$$998\,244\,353$$$.

Examples

Input

1 0011 0101

Output

1

Input

1 0011 0110

Output

0

Input

1 0??1 01??

Output

2

Input

2 000????? 01010101

Output

3

Input

2 0??????? 1???????

Output

68

Input

5 0101010101010101010101010101010101010101010101010101010101010101 ????????????????????????????????????????????????????????????????

Output

935297567

Note

In the first example, we can check that the condition $$$h(s, c) \ge 2^k$$$ for all cyclic shift $$$c$$$ of $$$t$$$ is satisfied. In particular:

- for $$$c = \mathtt{0101}$$$, $$$h(s, c) = h(\mathtt{0110}, \mathtt{0101}) = 2 \ge 2^1$$$;
- for $$$c = \mathtt{1010}$$$, $$$h(s, c) = h(\mathtt{0110}, \mathtt{1010}) = 2 \ge 2^1$$$.

In the second example, there exists a cycle shift $$$c$$$ of $$$t$$$ such that $$$h(s, c) < 2^k$$$ (in particular, $$$c = \mathtt{0011}$$$, and $$$h(s, c) = h(\mathtt{0011}, \mathtt{0011}) = 0$$$).

In the third example, there are $$$2$$$ possible ways to recover the missing characters:

- $$$s = \mathtt{0101}$$$, $$$t = \mathtt{0110}$$$;
- $$$s = \mathtt{0011}$$$, $$$t = \mathtt{0101}$$$.

In the fourth example, there are $$$3$$$ possible ways to recover the missing characters:

- $$$s = \mathtt{00011110}$$$, $$$t = \mathtt{01010101}$$$;
- $$$s = \mathtt{00011011}$$$, $$$t = \mathtt{01010101}$$$;
- $$$s = \mathtt{00001111}$$$, $$$t = \mathtt{01010101}$$$.

Codeforces (c) Copyright 2010-2024 Mike Mirzayanov

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