<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
        valid = {}
        for mask in range(3**m):
            color = []
            mm = mask
            for i in range(m):
                color.append(mm % 3)
                mm //= 3
            if any(color[i] == color[i + 1] for i in range(0, m - 1)):
                continue
            valid[mask] = color
        
        adjacent = {}
        for mask1, color1 in valid.items():
            adjacent[mask1] = []
            for mask2, color2 in valid.items():
                if any(x == y for x, y in zip(color1, color2)):
                    continue
                adjacent[mask1].append(mask2)

        f = [int(mask in valid) for mask in range(3**m)]
        for i in range(1, n):
            g = [0] * (3**m)
            for mask2 in valid.keys(): 
                for mask1 in adjacent[mask2]:
                    g[mask2] += f[mask1]
                if g[mask2] >= mod:
                        g[mask2] -= mod
            f = g

        return sum(f) % mod

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