<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>

INF = float('inf')

class Solution:
    def minimumSum(self, grid: List[List[int]]) -> int:
        m, n = len(grid), len(grid[0])

        def f1(row_lo, row_hi, col_lo, col_hi):
            imin, imax, jmin, jmax = INF, -INF, INF, -INF
            for i in range(row_lo, row_hi):
                for j in range(col_lo, col_hi):
                    if grid[i][j] == 1:
                        imin, imax, jmin, jmax = min(imin, i), max(
                            imax, i), min(jmin, j), max(jmax, j)

            return (imax - imin + 1) * (jmax - jmin + 1)

        def f2(row_lo, row_hi, col_lo, col_hi):
            res = INF
            for i in range(row_lo, row_hi):
                res = min(res, f1(row_lo, i, col_lo, col_hi) + f1(i, row_hi, col_lo, col_hi))

            for j in range(col_lo, col_hi):
                res = min(res, f1(row_lo, row_hi, col_lo, j) + f1(row_lo, row_hi, j, col_hi))

            return res
        
        res = INF
        for i in range(1, m):
            res = min(res, f1(0, i, 0, n) + f2(i, m, 0, n), f2(0, i, 0, n) + f1(i, m, 0, n))
        
        for j in range(1, n):
            res = min(res, f1(0, m, 0, j) + f2(0, m, j, n), f2(0, m, 0, j) + f1(0, m, j, n))
        
        return res

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