<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>

import itertools
from typing import Callable, List

import numpy as np
def tree_search(
    start: int,
    goal: int,
    adjacent: Callable[[int, int], bool],
    n_nodes: int,
) -> List[int]:

    unexplored = list(range(n_nodes))

    lst = [start]
    while lst:
        node = lst.pop()
        if node == goal:
            return True
        else:
            new_unexplored, new_nodes = [], []
            for i in unexplored:
                new_nodes.append(i) if adjacent(node, i) else new_unexplored.append(i)
            lst += new_nodes
            unexplored = new_unexplored

    return False

class Solution:
    def canReachCorner(self, X: int, Y: int, circles: List[List[int]]) -> bool:

        def adjacent(i: int, j: int):
            if i < 4 and j < 4:
                return False
            if i >= 4 and j >= 4:
                # two circles
                x1, y1, r1 = circles[i - 4]
                x2, y2, r2 = circles[j - 4]
                dist = ((x1 - x2) ** 2 + (y1 - y2) ** 2) ** 0.5 - (r1 + r2)
                return dist <= 0
            else:
                x, y, r = circles[max(i, j) - 4]
                if min(i, j) == 0:
                    return r >= y
                if min(i, j) == 1:
                    return r >= X - x
                if min(i, j) == 2:
                    return r >= Y - y
                if min(i, j) == 3:
                    return r >= x

        connected = False
        if tree_search(0, 3, adjacent, n_nodes=len(circles) + 4):
            connected = True
            return False
        if tree_search(1, 2, adjacent, n_nodes=len(circles) + 4):
            connected = True
            return False
        if tree_search(0, 2, adjacent, n_nodes=len(circles) + 4):
            connected = True
            return False
        if tree_search(1, 3, adjacent, n_nodes=len(circles) + 4):
            connected = True
            return False

        return not connected

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