<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>

from collections import defaultdict
from math import sqrt

class Solution:
    def canReachCorner(self, X: int, Y: int, circles: List[List[int]]) -> bool:
        def helper(c1, c2):
            x1, y1, r1 = c1
            x2, y2, r2 = c2
            R = sqrt((x1 - x2) ** 2 + (y1 - y2) ** 2)
            if R < 10 ** -6:
                return False
            dis = 2 * (r1 ** 2 + r2 ** 2) / R / R - (r1 ** 2 - r2 ** 2) ** 2 / (R ** 4) - 1
            if dis < 0:
                return False
            for sign in [-1, 1]:
                x = (x1 + x2) / 2 + (r1 ** 2 - r2 ** 2) / 2 / R / R * (x2 - x1) + sign * sqrt(dis) * (y2 - y1) / 2
                y = (y1 + y2) / 2 + (r1 ** 2 - r2 ** 2) / 2 / R / R * (y2 - y1) - sign * sqrt(dis) * (x2 - x1) / 2
                if 0 <= x < X and 0 <= y < Y:
                    return True
            return False
        adj = defaultdict(list)
        for i in range(len(circles)):
            for j in range(i + 1, len(circles)):
                c1, c2 = circles[i], circles[j]
                if helper(c1, c2):
                    adj[i].append(j)
                    adj[j].append(i)
        def DFS(node):
            visited.add(node)
            c = circles[node]
            if (c[1] - c[2] <= 0 < c[1] + c[2] and 0 <= c[0] <= X) or (c[0] - c[2] < X <= c[0] + c[2] and 0 <= c[1] <= Y):
                return True
            for neigh in adj[node]:
                if neigh in visited:
                    continue
                valid = DFS(neigh)
                if valid:
                    return True
            return False
        visited = set()
        for i in range(len(circles)):
            if i in visited:
                continue
            c = circles[i]
            if (c[0] - c[2] <= 0 < c[0] + c[2] and 0 <= c[1] <= Y) or (c[1] - c[2] < Y <= c[1] + c[2] and 0 <= c[0] <= X):
                valid = DFS(i)
                if valid:
                    return False
        return True
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