<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:
        g = defaultdict(list)
        n = len(circles)
        for i in range(n - 1):
            c1 = circles[i]
            if c1[0] < -c1[2] or c1[0] > X + c1[2] or c1[1] < -c1[2] or c1[1] > Y + c1[2]:
                continue
            for j in range(i + 1, n):
                c2 = circles[j]
                if c2[0] < -c2[2] or c2[0] > X + c2[2] or c2[1] < -c2[2] or c2[1] > Y + c2[2]:
                    continue
                if (c1[0] - c2[0]) ** 2 + (c1[1] - c2[1]) ** 2 <= (c1[2] + c2[2]) ** 2:
                    g[i].append(j)
                    g[j].append(i)
        grouped = set()
        for i in range(n):
            if i in grouped:
                continue
            que = deque()
            visited = set()
            que.append(i)
            while que:
                curr = que.popleft()
                if curr in visited:
                    continue
                visited.add(curr)
                if curr not in g:
                    continue
                for nextC in g[curr]:
                    que.append(nextC)
            left, right, up, down = False, False, False, False
            for circleId in visited:
                c = circles[circleId]
                if c[0] - c[2] <= 0 and (sqrt(c[2]**2 - c[0]**2) + c[1] >= 0 or -(sqrt(c[2]**2 - c[0]**2)) + c[1] <= Y):
                    left = True
                if c[0] + c[2] >= X and (sqrt(c[2]**2 - (X - c[0])**2) + c[1] >= 0 or -(sqrt(c[2]**2 - (X - c[0])**2)) + c[1] <= Y):
                    right = True
                if c[1] - c[2] <= 0 and (sqrt(c[2]**2 - c[1]**2) + c[0] >= 0 or -(sqrt(c[2]**2 - c[1]**2)) + c[0] <= Y):
                    down = True
                if c[1] + c[2] >= Y and (sqrt(c[2]**2 - (Y - c[1])**2) + c[0] >= 0 or -(sqrt(c[2]**2 - (Y - c[1])**2)) + c[0] <= Y):
                    up = True
                grouped.add(circleId)
                if (left and down) or (up and right) or (left and right) or (up and down):
                    return False
        return True
                    
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