<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>



class Solution:

    grid = []

    @cache
    def explore(this, loc, depth):
        u,v = loc

        m = len(this.grid) - 1 
        n = len(this.grid[0]) - 1

        if (u>m) or (v>n):
            return False

        depth += this.grid[u][v]

        if (u,v) == (m,n):
            if (depth==0): return True

        if 0<=depth<=(m-u+n-v):
            return this.explore((u+1,v),depth) or this.explore((u,v+1),depth)
        else:
            return False
    

    def hasValidPath(this, grid: List[List[str]]) -> bool:
        this.grid = [ [ 1 if c=='(' else -1 for c in r ] for r in grid ]

        return this.explore((0,0),0)

    
 
        

        
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