<p>You are given two positive integers <code>xCorner</code> and <code>yCorner</code>, and a 2D array <code>circles</code>, where <code>circles[i] = [x<sub>i</sub>, y<sub>i</sub>, r<sub>i</sub>]</code> denotes a circle with center at <code>(x<sub>i</sub>, y<sub>i</sub>)</code> and radius <code>r<sub>i</sub></code>.</p>

<p>There is a rectangle in the coordinate plane with its bottom left corner at the origin and top right corner at the coordinate <code>(xCorner, yCorner)</code>. You need to check whether there is a path from the bottom left corner to the top right corner such that the <strong>entire path</strong> lies inside the rectangle, <strong>does not</strong> touch or lie inside <strong>any</strong> circle, and touches the rectangle <strong>only</strong> at the two corners.</p>

<p>Return <code>true</code> if such a path exists, and <code>false</code> otherwise.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">xCorner = 3, yCorner = 4, circles = [[2,1,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">true</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/05/18/example2circle1.png" style="width: 346px; height: 264px;" /></p>

<p>The black curve shows a possible path between <code>(0, 0)</code> and <code>(3, 4)</code>.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">xCorner = 3, yCorner = 3, circles = [[1,1,2]]</span></p>

<p><strong>Output:</strong> <span class="example-io">false</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/05/18/example1circle.png" style="width: 346px; height: 264px;" /></p>

<p>No path exists from <code>(0, 0)</code> to <code>(3, 3)</code>.</p>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">xCorner = 3, yCorner = 3, circles = [[2,1,1],[1,2,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">false</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/05/18/example0circle.png" style="width: 346px; height: 264px;" /></p>

<p>No path exists from <code>(0, 0)</code> to <code>(3, 3)</code>.</p>
</div>

<p><strong class="example">Example 4:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">xCorner = 4, yCorner = 4, circles = [[5,5,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">true</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/08/04/rectangles.png" style="width: 346px; height: 264px;" /></p>
</div>

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

<ul>
	<li><code>3 &lt;= xCorner, yCorner &lt;= 10<sup>9</sup></code></li>
	<li><code>1 &lt;= circles.length &lt;= 1000</code></li>
	<li><code>circles[i].length == 3</code></li>
	<li><code>1 &lt;= x<sub>i</sub>, y<sub>i</sub>, r<sub>i</sub> &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def canReachCorner(self, X: int, Y: int, circles: List[List[int]]) -> bool:
        def does_affect(x, y, r):
            if (
                (x > X and x - X >= r) or
                (y > Y and y - Y >= r)
            ):
                return False
            return True

        circles = [circle for circle in circles if does_affect(*circle)]

        border_intersection = {border: [0] * len(circles) for border in ("b", "t", "r", "l")}
        circle_intersection = [set() for _ in range(len(circles))]
        for i in range(len(circles)):
            x, y, r = circles[i]

            if x <= r:
                border_intersection["l"][i] = 1
            if y <= r:
                border_intersection["b"][i] = 1
            if abs(x - X) <= r:
                border_intersection["r"][i] = 1
            if abs(y - Y) <= r:
                border_intersection["t"][i] = 1

            for j in range(i + 1, len(circles)):
                x_, y_, r_ = circles[j]
                if (x - x_)**2 + (y - y_)**2 <= (r + r_)**2:
                    circle_intersection[i].add(j)
                    circle_intersection[j].add(i)

        def bfs(circle):
            cur = [circle]
            res = set()
            while cur:
                nxt = []
                for circle in cur:
                    for border in ("b", "t", "r", "l"):
                        if border_intersection[border][circle]:
                            res.add(border)
                    for nxt_circle in circle_intersection[circle]:
                        if nxt_circle in circles:
                            circles.remove(nxt_circle)
                            nxt.append(nxt_circle)
                cur = nxt
            return res

        circles = set(range(len(circles)))
        while circles:
            circle = circles.pop()
            res = bfs(circle)
            if (
                "l" in res and "b" in res or
                "l" in res and "r" in res or
                "t" in res and "b" in res or
                "t" in res and "r" in res
            ):
                return False

        return True

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