<p>You are given a 2D <strong>binary</strong> array <code>grid</code>. You need to find 3 <strong>non-overlapping</strong> rectangles having <strong>non-zero</strong> areas with horizontal and vertical sides such that all the 1&#39;s in <code>grid</code> lie inside these rectangles.</p>

<p>Return the <strong>minimum</strong> possible sum of the area of these rectangles.</p>

<p><strong>Note</strong> that the rectangles are allowed to touch.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">grid = [[1,0,1],[1,1,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">5</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/05/14/example0rect21.png" style="padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; width: 280px; height: 198px;" /></p>

<ul>
	<li>The 1&#39;s at <code>(0, 0)</code> and <code>(1, 0)</code> are covered by a rectangle of area 2.</li>
	<li>The 1&#39;s at <code>(0, 2)</code> and <code>(1, 2)</code> are covered by a rectangle of area 2.</li>
	<li>The 1 at <code>(1, 1)</code> is covered by a rectangle of area 1.</li>
</ul>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">grid = [[1,0,1,0],[0,1,0,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">5</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/05/14/example1rect2.png" style="padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; width: 356px; height: 198px;" /></p>

<ul>
	<li>The 1&#39;s at <code>(0, 0)</code> and <code>(0, 2)</code> are covered by a rectangle of area 3.</li>
	<li>The 1 at <code>(1, 1)</code> is covered by a rectangle of area 1.</li>
	<li>The 1 at <code>(1, 3)</code> is covered by a rectangle of area 1.</li>
</ul>
</div>

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

<ul>
	<li><code>1 &lt;= grid.length, grid[i].length &lt;= 30</code></li>
	<li><code>grid[i][j]</code> is either 0 or 1.</li>
	<li>The input is generated such that there are at least three 1&#39;s in <code>grid</code>.</li>
</ul>

class Solution:
    def minimumSum(self, grid: List[List[int]]) -> int:

        @lru_cache(None)
        def getArea(startRow: int, endRow: int, startCol: int, endCol: int) -> int:

            if endRow < startRow or endCol < startCol:
                return inf
            maxRow = maxCol = -inf
            minRow = minCol = inf
            for row in range(startRow, endRow + 1):
                for col in range(startCol, endCol + 1):
                    if grid[row][col] == 1:
                        if row < minRow: minRow = row
                        if row > maxRow: maxRow = row
                        if col < minCol: minCol = col
                        if col > maxCol: maxCol = col
            if minRow == inf or minCol == inf or maxRow == -inf or maxCol == -inf:
                return 0
            return (maxRow - minRow + 1) * (maxCol - minCol + 1)


        ans, numRows, numCols = inf, len(grid), len(grid[0])

        for row, col in product(range(numRows), range(numCols)):

            ans = min(ans, getArea(*(0, row), *(0, numCols - 1)) + 
                            getArea(*(row + 1, numRows - 1), *(0, col)) + 
                            getArea(*(row + 1, numRows - 1), *(col + 1, numCols - 1))  )

            ans = min(ans, getArea(0, row, 0, col) + 
                            getArea(0, row, col + 1, numCols - 1) + 
                            getArea(row + 1, numRows - 1, 0, numCols - 1))

            ans = min(ans, getArea(0, numRows - 1, 0, col) + 
                            getArea(0, row, col + 1, numCols - 1) + 
                            getArea(row + 1, numRows - 1, col + 1, numCols - 1))

            ans = min(ans, getArea(0, numRows - 1, col + 1, numCols - 1) + 
                            getArea(0, row, 0, col) + 
                            getArea(row + 1, numRows - 1, 0, col))


        for upperRow in range(numRows):
            for lowerRow in range(upperRow, numRows):
                ans = min(ans, getArea(0, upperRow, 0, numCols - 1) + getArea(upperRow + 1, lowerRow, 0, numCols - 1) + getArea(lowerRow + 1, numRows - 1, 0, numCols - 1))
        for leftCol in range(numCols):
            for rightCol in range(leftCol, numCols):
                ans = min(ans, getArea(0, numRows - 1, 0, leftCol) + getArea(0, numRows - 1, leftCol + 1, rightCol) + getArea(0, numRows - 1, rightCol + 1, numCols - 1))
        return ans
742