<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:
        m, n = len(grid), len(grid[0])
        res = math.inf

        def min_area(row_i, row_j, col_i, col_j) -> int:
            min_r = min_c = math.inf
            max_r = max_c = 0
            for i in range(row_i, row_j + 1):
                for j in range(col_i, col_j + 1):
                    if grid[i][j] == 1:
                        if math.isinf(min_r):
                            min_r = i
                        min_c = min(min_c, j)
                        max_r = i
                        max_c = max(max_c, j)
            return (max_c - min_c + 1) * (max_r - min_r + 1)

        for i in range(m):

            top = min_area(0, i, 0, n - 1)

            for j in range(n):
                down_left = min_area(i + 1, m - 1, 0, j)
                down_right = min_area(i + 1, m - 1, j + 1, n - 1)
                res = min(res, top + down_left + down_right)

            for j in range(i + 1, m):
                mid = min_area(i + 1, j, 0, n - 1)
                down = min_area(j + 1, m - 1, 0, n - 1)
                res = min(res, top + mid + down)

            down = min_area(i + 1, m - 1, 0, n - 1)
            for j in range(n):
                top_left = min_area(0, i, 0, j)
                top_right = min_area(0, i, j + 1, n - 1)
                res = min(res, top_left + top_right + down)

        for j in range(n):
            left = min_area(0, m - 1, 0, j)
            for i in range(m):
                right_top = min_area(0, i, j + 1, n - 1)
                right_down = min_area(i + 1, m - 1, j + 1, n - 1)
                res = min(res, left + right_top + right_down)
            
            right = min_area(0, m - 1, j + 1, n - 1)
            for i in range(m):
                left_top = min_area(0, i, 0, j)
                left_down = min_area(i + 1, m - 1, 0, j)
                res = min(res, right + left_top + left_down)

            for i in range(j + 1, n):
                mid = min_area(0, m - 1, j + 1, i)
                right = min_area(0, m - 1, i + 1, n - 1)
                res = min(res, left + mid + right)

        return res

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