<p>There is a strange printer with the following two special requirements:</p>

<ul>
	<li>On each turn, the printer will print a solid rectangular pattern of a single color on the grid. This will cover up the existing colors in the rectangle.</li>
	<li>Once the printer has used a color for the above operation, <strong>the same color cannot be used again</strong>.</li>
</ul>

<p>You are given a <code>m x n</code> matrix <code>targetGrid</code>, where <code>targetGrid[row][col]</code> is the color in the position <code>(row, col)</code> of the grid.</p>

<p>Return <code>true</code><em> if it is possible to print the matrix </em><code>targetGrid</code><em>,</em><em> otherwise, return </em><code>false</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/12/23/print1.jpg" style="width: 600px; height: 175px;" />
<pre>
<strong>Input:</strong> targetGrid = [[1,1,1,1],[1,2,2,1],[1,2,2,1],[1,1,1,1]]
<strong>Output:</strong> true
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/12/23/print2.jpg" style="width: 600px; height: 367px;" />
<pre>
<strong>Input:</strong> targetGrid = [[1,1,1,1],[1,1,3,3],[1,1,3,4],[5,5,1,4]]
<strong>Output:</strong> true
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> targetGrid = [[1,2,1],[2,1,2],[1,2,1]]
<strong>Output:</strong> false
<strong>Explanation:</strong> It is impossible to form targetGrid because it is not allowed to print the same color in different turns.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == targetGrid.length</code></li>
	<li><code>n == targetGrid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 60</code></li>
	<li><code>1 &lt;= targetGrid[row][col] &lt;= 60</code></li>
</ul>

class Solution:
    def isPrintable(self, targetGrid: List[List[int]]) -> bool:

        def dp(color: int) -> bool:

            rows, cols = list(zip(*colors[color]))

            mn_r, mx_r, mn_c, mx_c = (min(rows), max(rows) + 1,
                                      min(cols), max(cols) + 1)
            
            for i, j in product(range(mn_r, mx_r), range(mn_c, mx_c)):
                check = targetGrid[i][j] > 0 and targetGrid[i][j] != color
                if check: return False

            for i, j in product(range(mn_r, mx_r), range(mn_c, mx_c)):
                targetGrid[i][j] = 0

            return True


        m, n = len(targetGrid), len(targetGrid[0])
        colors = defaultdict(list)

        for i, j in product(range(m), range(n)):
            colors[targetGrid[i][j]].append((i,j))

        color_set = set(colors)

        while color_set:
            tmp = set(filter(lambda x: not dp(x), color_set))
            if color_set == tmp: return False
            color_set = tmp

        return True
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