<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 = 1000000007
        
        def solve_rel(dp, coeff):
            for _ in range(n - 1):
                for i in range(len(dp)):
                    tmp = 0
                    for j, c in enumerate(coeff[i]):
                        # If j >= i, use the last updated value; otherwise, use the second last.
                        tmp += c * dp[j][-1 if j >= i else -2]
                    dp[i].append(tmp % MOD)
            return sum(e[-1] for e in dp) % MOD

        if m == 1:
            return solve_rel([[3]], [[2]])
        if m == 2:
            return solve_rel([[6]], [[3]])
        elif m == 3:
            return solve_rel([[6], [6]], [[3, 2], [2, 2]])
        elif m == 4:
            return solve_rel(
                [[6], [6], [6], [6]], 
                [[3, 1, 2, 2], [1, 2, 1, 1], [2, 1, 2, 2], [2, 1, 2, 2]]
            )
        else:
            return solve_rel(
                [[6], [6], [6], [6], [6], [6], [6], [6]],
                [
                    [2, 1, 1, 1, 2, 2, 1, 1], 
                    [1, 2, 1, 1, 0, 0, 1, 0], 
                    [1, 1, 2, 1, 1, 1, 1, 1], 
                    [1, 1, 1, 2, 2, 2, 1, 1], 
                    [2, 0, 1, 2, 3, 2, 1, 2], 
                    [2, 0, 1, 2, 2, 2, 1, 2], 
                    [1, 1, 1, 1, 1, 1, 2, 1], 
                    [1, 0, 1, 1, 2, 2, 1, 2]
                ]
            )
17200