<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

        def generate_valid_rows(m):
            def backtrack(row, last_color):
                if len(row) == m:
                    valid_rows.append(tuple(row))
                    return
                for color in range(3):
                    if color != last_color:
                        row.append(color)
                        backtrack(row, color)
                        row.pop()

            valid_rows = []
            backtrack([], -1)
            return valid_rows

        def are_compatible(row1, row2):
            return all(c1 != c2 for c1, c2 in zip(row1, row2))

        valid_rows = generate_valid_rows(m)
        k = len(valid_rows)

        dp = [[0] * k for _ in range(n)]
        for j in range(k):
            dp[0][j] = 1

        for i in range(1, n):
            for j in range(k):
                for l in range(k):
                    if are_compatible(valid_rows[j], valid_rows[l]):
                        dp[i][j] = (dp[i][j] + dp[i-1][l]) % MOD

        return sum(dp[n-1]) % MOD
2898