Problem p03626 - Generation 3

Orig Description

Score : 400 points
Problem StatementWe have a board with a 2 \times N grid.
Snuke covered the board with N dominoes without overlaps.
Here, a domino can cover a 1 \times 2 or 2 \times 1 square.
Then, Snuke decided to paint these dominoes using three colors: red, cyan and green.
Two dominoes that are adjacent by side should be painted by different colors.
Here, it is not always necessary to use all three colors.
Find the number of such ways to paint the dominoes, modulo 1000000007.
The arrangement of the dominoes is given to you as two strings S_1 and S_2 in the following manner:
Each domino is represented by a different English letter (lowercase or uppercase).
The j-th character in S_i represents the domino that occupies the square at the i-th row from the top and j-th column from the left.
Constraints
1 \leq N \leq 52
|S_1| = |S_2| = N
S_1 and S_2 consist of lowercase and uppercase English letters.
S_1 and S_2 represent a valid arrangement of dominoes.
InputInput is given from Standard Input in the following format:
N
S_1
S_2
OutputPrint the number of such ways to paint the dominoes, modulo 1000000007.
Sample Input 13
aab
ccb
Sample Output 16
There are six ways as shown below:
Sample Input 21
Z
Z
Sample Output 23
Note that it is not always necessary to use all the colors.
Sample Input 352
RvvttdWIyyPPQFFZZssffEEkkaSSDKqcibbeYrhAljCCGGJppHHn
RLLwwdWIxxNNQUUXXVVMMooBBaggDKqcimmeYrhAljOOTTJuuzzn
Sample Output 3958681902

Extracted Specification

An integer N (1 ≤ N ≤ 52), representing some quantity or size.
A string S1 of length N, consisting of uppercase and lowercase English letters.
A string S2 of length N, consisting of uppercase and lowercase English letters.

### Example Input:

```
3
aab
ccb
```

### Function Signature:
Write a function f(N, S1, S2) that takes in the input.
def f(N: int, S1: str, S2: str):
    '''
    N: an integer
    S1: a string
    S2: a string
    '''

Test Case Generator

Sure! Here is a test case generator for the given problem statement about painting the dominoes on a 2xN grid:

```python
import random
import string

def generate_valid_domino_placement(N):
    letters = string.ascii_letters
    used_letters = set()
    
    def get_new_letter():
        letter = random.choice(letters)
        while letter in used_letters:
            letter = random.choice(letters)
        used_letters.add(letter)
        return letter
    
    S1 = []
    S2 = []
    i = 0
    while i < N:
        if random.choice([True, False]):  # Decide to place vertical or horizontal domino
            if i + 1 < N and (i == 0 or S1[i-1] != S1[i-2] and S2[i-1] != S2[i-2]):
                # Place a horizontal domino
                letter = get_new_letter()
                S1.extend([letter, letter])
                S2.extend([letter, letter])
                i += 2
            else:
                # Place a vertical domino
                letter = get_new_letter()
                S1.append(letter)
                S2.append(letter)
                i += 1
        else:
            # Place a vertical domino
            letter = get_new_letter()
            S1.append(letter)
            S2.append(letter)
            i += 1
    
    return ''.join(S1[:N]), ''.join(S2[:N])

def tcgen():
    N = random.randint(1, 52)
    S1, S2 = generate_valid_domino_placement(N)
    return N, S1, S2

# Example usage:
N, S1, S2 = tcgen()
print(N)
print(S1)
print(S2)
```

### Explanation:
- **Function `generate_valid_domino_placement(N)`**: This function generates two strings `S1` and `S2` that represent a valid arrangement of dominoes on a 2xN grid.
  - It ensures that each domino is represented by the same letter in adjacent cells.
  - It randomly decides whether to place the dominos vertically (occupying one cell in each row) or horizontally (occupying two adjacent cells in a row), ensuring a valid placement.
- **Function `tcgen()`**: This function generates a random test case:
  - It generates a random number \( N \) (1 ≤ \( N \) ≤ 52).
  - It generates the two strings `S1` and `S2` representing the domino placement.
  - It returns \( N \), `S1`, and `S2`.

This generator will ensure that the constraints are met and the input is valid for the problem statement.

Extract Arguments

def extract_arguments(fh):
    N = int(fh.readline().strip())
    S_1 = fh.readline().strip()
    S_2 = fh.readline().strip()
    return N, S_1, S_2