<p>A parentheses string is a <strong>non-empty</strong> string consisting only of <code>&#39;(&#39;</code> and <code>&#39;)&#39;</code>. It is <strong>valid</strong> if <strong>any</strong> of the following conditions is <strong>true</strong>:</p>

<ul>
	<li>It is <code>()</code>.</li>
	<li>It can be written as <code>AB</code> (<code>A</code> concatenated with <code>B</code>), where <code>A</code> and <code>B</code> are valid parentheses strings.</li>
	<li>It can be written as <code>(A)</code>, where <code>A</code> is a valid parentheses string.</li>
</ul>

<p>You are given an <code>m x n</code> matrix of parentheses <code>grid</code>. A <strong>valid parentheses string path</strong> in the grid is a path satisfying <strong>all</strong> of the following conditions:</p>

<ul>
	<li>The path starts from the upper left cell <code>(0, 0)</code>.</li>
	<li>The path ends at the bottom-right cell <code>(m - 1, n - 1)</code>.</li>
	<li>The path only ever moves <strong>down</strong> or <strong>right</strong>.</li>
	<li>The resulting parentheses string formed by the path is <strong>valid</strong>.</li>
</ul>

<p>Return <code>true</code> <em>if there exists a <strong>valid parentheses string path</strong> in the grid.</em> Otherwise, return <code>false</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/03/15/example1drawio.png" style="width: 521px; height: 300px;" />
<pre>
<strong>Input:</strong> grid = [[&quot;(&quot;,&quot;(&quot;,&quot;(&quot;],[&quot;)&quot;,&quot;(&quot;,&quot;)&quot;],[&quot;(&quot;,&quot;(&quot;,&quot;)&quot;],[&quot;(&quot;,&quot;(&quot;,&quot;)&quot;]]
<strong>Output:</strong> true
<strong>Explanation:</strong> The above diagram shows two possible paths that form valid parentheses strings.
The first path shown results in the valid parentheses string &quot;()(())&quot;.
The second path shown results in the valid parentheses string &quot;((()))&quot;.
Note that there may be other valid parentheses string paths.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/03/15/example2drawio.png" style="width: 165px; height: 165px;" />
<pre>
<strong>Input:</strong> grid = [[&quot;)&quot;,&quot;)&quot;],[&quot;(&quot;,&quot;(&quot;]]
<strong>Output:</strong> false
<strong>Explanation:</strong> The two possible paths form the parentheses strings &quot;))(&quot; and &quot;)((&quot;. Since neither of them are valid parentheses strings, we return false.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 100</code></li>
	<li><code>grid[i][j]</code> is either <code>&#39;(&#39;</code> or <code>&#39;)&#39;</code>.</li>
</ul>

from collections import deque
from dataclasses import dataclass
from functools import cache, partial
from typing import Literal, Sequence, TypeVar, assert_never

Coordinate = tuple[int, int]
T = TypeVar("T")
Parenthesis = Literal["(", ")"]


def get_cell_safe(grid: list[list[T]], pos: Coordinate):
    x, y = pos
    if x < 0 or y < 0 or y >= len(grid) or x >= len(row := grid[y]):
        return None
    return row[x]


class Solution:
    def hasValidPath(self, grid: list[list[Parenthesis]]) -> bool:
        gx = len(grid[0]) - 1
        gy = len(grid) - 1
        get_field = partial(get_cell_safe, grid)

        @cache
        def find_path(x: int, y: int, opened: int):
            if x == gx and y == gy and opened == 0:
                return True
            for neighbor in ((x, y + 1), (x + 1, y)):
                if (field := get_field(neighbor)) is None:
                    continue
                if (neighbor_opened := opened + (1 if field == "(" else -1)) < 0:
                    continue
                if find_path(neighbor[0], neighbor[1], neighbor_opened):
                    return True
            return False

        if get_field((0, 0)) != "(":
            return False
        return find_path(0, 0, 1)

    def __repr__(self):
        return f"{self.__class__.__name__}()"

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