<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:

        def all_possible_columns(columns = []):
            if len(columns) == m:
                return [columns]
            results = []
            for c in range(3):
                if not columns or columns[-1] != c:
                    results.extend(all_possible_columns(columns + [c]))
            return results
        def match_all_columns(arr):
            def test_columns(c1, c2):
                return all(a != b for a, b in zip(c1, c2))
            n = len(arr)
            result = [[] for _ in range(n)]
            for i in range(n):
                for j in range(i + 1, n):
                    if test_columns(arr[i], arr[j]):
                        result[i].append(j)
                        result[j].append(i)
            return result

        all_columns = all_possible_columns()
        all_good = match_all_columns(all_columns)
        @cache
        def search(idx, depth):
            if depth == n:
                return 1
            tot = 0
            for nei in all_good[idx]:
                tot += search(nei, depth + 1)
            return tot % 1000000007

            
        ans = 0
        for i in range(len(all_columns)):
            ans += search(i, 1)

        return ans % 1000000007



        
411