<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 dist(x1, y1, x2, y2):
            return ((x1 - x2)**2 + (y1 - y2)**2)**.5

        def intersecting(x1, y1, r1, x2, y2, r2):
            return dist(x1, y1, x2, y2) <= r1 + r2
        
        def within(x, y, i):
            return intersecting(x, y, 0, circles[i][0], circles[i][1], circles[i][2])

        for i in range(len(circles)):
            if within(0, 0, i) or within(X, Y, i):
                return False

        g = defaultdict(list)
        for i, (x1, y1, r1) in enumerate(circles):
            for j, (x2, y2, r2) in enumerate(circles[:i]):
                if intersecting(x1, y1, r1, x2, y2, r2):
                    g[i].append(j)
                    g[j].append(i)
        
        def search(sources, predicate):
            q = deque()
            seen = set()
            for s in sources:
                if predicate(s):
                    return True
                q.append(s)
            while q:
                node = q.popleft()
                if node not in seen:
                    seen.add(node)
                    if predicate(node):
                        return True
                    for nxt in g.get(node, []):
                        if nxt not in seen:
                            q.append(nxt)
            return False

        
        ix0 = []
        for i in range(len(circles)):
            if circles[i][0] <= circles[i][2]:
                ix0.append(i)

        if search(ix0, lambda n: circles[n][1] <= circles[n][2]):
            return False

        if search(ix0, lambda n: abs(X - circles[n][0]) <= circles[n][2]):
            return False

        iyY = []
        for i in range(len(circles)):
            if abs(Y - circles[i][1]) <= circles[i][2]:
                iyY.append(i)

        if search(iyY, lambda n: circles[n][1] <= circles[n][2]):
            return False

        if search(iyY, lambda n: abs(X - circles[n][0]) <= circles[n][2]):
            return False

        return True
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