<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>


from functools import cache, lru_cache
from itertools import product


class Solution:
    def colorTheGrid(self, m: int, n: int) -> int:
        mod = 10 ** 9 + 7

        def check(pos):
            return all(a != b for a, b in zip(pos, pos[1:]))

        def neighs(pos1, pos2):
            return all(a != b for a, b in zip(pos1, pos2))

        states = {''.join(pos) for pos in product('RGB', repeat=m) if check(pos)}
        adj = {}
        for state in states:
            adj[state] = [next_state for next_state in states if neighs(state, next_state)]

        @cache
        def f(prv_state, N):
            if N == 0:
                return 1
            return sum(f(next_state, N - 1) for next_state in adj[prv_state]) % mod

        return sum(f(state, n - 1) for state in states) % mod

577