<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:
        n = len(grid)
        m = len(grid[0])
        
        def get_area(rect):
            a, b, c, d = rect
            x1, y1, x2, y2 = inf,inf, -1, -1
            for i in range(a, c + 1):
                for j in range(b, d + 1):
                    if grid[i][j]:
                        x1 = min(x1, i)
                        y1 = min(y1, j)
                        x2 = max(x2, i)
                        y2 = max(y2, j)
            if x1 ==inf:
                return 0
            return (x2 - x1 + 1) * (y2 - y1 + 1)
        
        def iter_splits(rect):
            x1, y1, x2, y2 = rect
            for i in range(x1, x2):
                rect1 = (x1, y1, i, y2)
                rect2 = (i + 1, y1, x2, y2)
                yield rect1, rect2
            for j in range(y1, y2):
                rect1 = (x1, y1, x2, j)
                rect2 = (x1, j + 1, x2, y2)
                yield rect1, rect2
        
        def solve(rect, splits=2):
            if splits == 0:
                return get_area(rect)
            ans =inf
            for rect1, rect2 in iter_splits(rect):
                for i in range(splits):
                    ans = min(ans, solve(rect1, i) + solve(rect2, splits -1 - i))
            return ans
        
        return solve((0, 0, n - 1, m - 1))
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