<p>You are given two integers <code>m</code> and <code>n</code>. Consider an <code>m x n</code> grid where each cell is initially white. You can paint each cell <strong>red</strong>, <strong>green</strong>, or <strong>blue</strong>. All cells <strong>must</strong> be painted.</p>

<p>Return<em> the number of ways to color the grid with <strong>no two adjacent cells having the same color</strong></em>. Since the answer can be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/22/colorthegrid.png" style="width: 200px; height: 50px;" />
<pre>
<strong>Input:</strong> m = 1, n = 1
<strong>Output:</strong> 3
<strong>Explanation:</strong> The three possible colorings are shown in the image above.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/22/copy-of-colorthegrid.png" style="width: 321px; height: 121px;" />
<pre>
<strong>Input:</strong> m = 1, n = 2
<strong>Output:</strong> 6
<strong>Explanation:</strong> The six possible colorings are shown in the image above.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> m = 5, n = 5
<strong>Output:</strong> 580986
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= m &lt;= 5</code></li>
	<li><code>1 &lt;= n &lt;= 1000</code></li>
</ul>

class Solution:
    def colorTheGrid(self, m: int, n: int) -> int:
        MOD = 10**9 + 7
        
        # Define a function to check if two columns are compatible (no adjacent cells have the same color)
        def compatible(col1, col2):
            for i in range(m):
                if col1[i] == col2[i]:
                    return False
            return True

        # Generate all possible colorings for a single column
        def generate_colorings(col, pos=0):
            if pos == m:
                return [col]
            colorings = []
            for color in range(3):
                if pos > 0 and col[pos-1] == color:
                    continue
                colorings.extend(generate_colorings(col + [color], pos + 1))
            return colorings

        # Generate all possible colorings for a single column
        all_colorings = generate_colorings([])
        
        # Create a dictionary to store the number of ways to color each column
        colorings_count = {tuple(col): 1 for col in all_colorings}

        # Iterate over each column from 1 to n
        for _ in range(n - 1):
            next_colorings_count = {}
            for col1 in all_colorings:
                for col2 in all_colorings:
                    if compatible(col1, col2):
                        next_col = tuple(col2)
                        if next_col in next_colorings_count:
                            next_colorings_count[next_col] += colorings_count[tuple(col1)]
                        else:
                            next_colorings_count[next_col] = colorings_count[tuple(col1)]
            colorings_count = next_colorings_count

        return sum(colorings_count.values()) % MOD

# Example usage:
solution = Solution()
print(solution.colorTheGrid(1, 1))  # Output: 3
print(solution.colorTheGrid(1, 2))  # Output: 6
print(solution.colorTheGrid(5, 5))  # Output: 580986
2235