<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 deque
import math
class Solution:
    def canReachCorner(self, X: int, Y: int, circles: List[List[int]]) -> bool:
        def fun( g, i, t1, t2, node):
            qu = deque([i])
            visit= [0]*(node + 4)
            visit[i] = 1
            while qu:
                x = qu.popleft()
                for y in g[x]:
                    if not visit[y]:
                        qu.append(y)
                        visit[y] = 1
            return visit[t1] or visit[t2]




        flag = 0
        temp = [((i, j), r) for i, j, r in circles]
        gf = [[] for _ in range(len(circles) + 4)]
        for i,((x, y), r) in enumerate(temp):
            if y <= r:
                gf[len(circles) + 1].append(i)
                gf[i].append(len(circles) + 1)
            if Y - y <= r:
                gf[len(circles) + 3].append(i)
                gf[i].append(len(circles) + 3)
            if x <= r:
                gf[len(circles)].append(i)
                gf[i].append(len(circles))
            if X - x <= r:
                gf[len(circles) + 2].append(i)
                gf[i].append(len(circles) + 2)
            for j in range(i + 1, len(circles)):
                a,b = temp[j][0]
                if r+temp[j][1] >= math.sqrt((x - a)**2 + (y - b)**2):
                    gf[i].append(j)
                    gf[j].append(i)
        flag = max(flag, fun(gf, len(circles), len(circles) + 1, len(circles) +2,len(circles) +4))  
        flag = max(flag, fun(gf, len(circles)+3,len(circles)+ 2, len(circles)+ 1, len(circles)+ 4)) 
        if flag:
            return False
        return True
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