<p>Given an array <code>rectangles</code> where <code>rectangles[i] = [x<sub>i</sub>, y<sub>i</sub>, a<sub>i</sub>, b<sub>i</sub>]</code> represents an axis-aligned rectangle. The bottom-left point of the rectangle is <code>(x<sub>i</sub>, y<sub>i</sub>)</code> and the top-right point of it is <code>(a<sub>i</sub>, b<sub>i</sub>)</code>.</p>

<p>Return <code>true</code> <em>if all the rectangles together form an exact cover of a rectangular region</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/03/27/perectrec1-plane.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> rectangles = [[1,1,3,3],[3,1,4,2],[3,2,4,4],[1,3,2,4],[2,3,3,4]]
<strong>Output:</strong> true
<strong>Explanation:</strong> All 5 rectangles together form an exact cover of a rectangular region.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/03/27/perfectrec2-plane.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> rectangles = [[1,1,2,3],[1,3,2,4],[3,1,4,2],[3,2,4,4]]
<strong>Output:</strong> false
<strong>Explanation:</strong> Because there is a gap between the two rectangular regions.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/03/27/perfecrrec4-plane.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> rectangles = [[1,1,3,3],[3,1,4,2],[1,3,2,4],[2,2,4,4]]
<strong>Output:</strong> false
<strong>Explanation:</strong> Because two of the rectangles overlap with each other.
</pre>

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

<ul>
	<li><code>1 &lt;= rectangles.length &lt;= 2 * 10<sup>4</sup></code></li>
	<li><code>rectangles[i].length == 4</code></li>
	<li><code>-10<sup>5</sup> &lt;= x<sub>i</sub> &lt; a<sub>i</sub> &lt;= 10<sup>5</sup></code></li>
	<li><code>-10<sup>5</sup> &lt;= y<sub>i</sub> &lt; b<sub>i</sub> &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution:
    def isRectangleCover(self, rectangles: List[List[int]]) -> bool:
        area_sum = 0
        mn_x = float("inf")
        mx_x = -float("inf")
        mn_y = float("inf")
        mx_y = -float("inf")
        seen_corners = {}
        seen_rects = set()

        for first_x, first_y, second_x, second_y in rectangles:
            area_sum += (second_x - first_x) * (second_y - first_y)
            mx_x = max(mx_x, second_x)
            mn_x = min(mn_x, first_x)
            mx_y = max(mx_y, second_y)
            mn_y = min(mn_y, first_y)

            if (first_x, first_y, second_x, second_y) in seen_rects:
                return False
            seen_rects.add((first_x, first_y, second_x, second_y))

            seen_corners[(first_x, first_y)] = seen_corners.get((first_x, first_y), 0) + 1
            seen_corners[(second_x, second_y)] = seen_corners.get((second_x, second_y), 0) + 1 
            seen_corners[(second_x, first_y)] = seen_corners.get((second_x, first_y), 0) + 1 
            seen_corners[(first_x, second_y)] = seen_corners.get((first_x, second_y), 0) + 1 


        main_corners = ((mn_x, mn_y), (mn_x, mx_y), (mx_x, mx_y), (mx_x, mn_y))
        for coors, count in seen_corners.items():
            if (coors[0], coors[1]) not in main_corners and count & 1 == 1: 
                return False

        return area_sum == (mx_x - mn_x) * (mx_y - mn_y)
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