<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:
        # Initialize boundaries
        left = bottom = float('inf')
        right = top = float('-inf')
        
        # Calculate the boundaries of the large rectangle
        for rectangle in rectangles:
            x, y, a, b = rectangle
            left = min(left, x)
            bottom = min(bottom, y)
            right = max(right, a)
            top = max(top, b)  # Corrected this line

        # Calculate the expected area of the large rectangle
        expectedArea = (right - left) * (top - bottom)

        # Calculate the actual total area of all small rectangles
        actualArea = 0
        for rectangle in rectangles:
            x, y, a, b = rectangle
            actualArea += (a - x) * (b - y)
        
        # Check if the total area matches and if there are no overlaps
        if expectedArea != actualArea:
            return False

        # To ensure no overlaps, use a set to track the vertices
        vertices = set()
        
        # Function to toggle vertex in the set
        def toggle_vertex(x, y):
            if (x, y) in vertices:
                vertices.remove((x, y))
            else:
                vertices.add((x, y))

        # Add or remove corners of each rectangle
        for rectangle in rectangles:
            x, y, a, b = rectangle
            toggle_vertex(x, y)
            toggle_vertex(x, b)
            toggle_vertex(a, y)
            toggle_vertex(a, b)

        # Check that the final set of vertices contains exactly the four corners of the large rectangle
        if len(vertices) != 4 or (left, bottom) not in vertices or (left, top) not in vertices or (right, bottom) not in vertices or (right, top) not in vertices:
            return False
        
        return True
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