You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in c++17.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```cpp
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
This 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 Format
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 Format
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:

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

Now solve the problem and return the code.