<p>Given a string <code>s</code> containing only three types of characters: <code>&#39;(&#39;</code>, <code>&#39;)&#39;</code> and <code>&#39;*&#39;</code>, return <code>true</code> <em>if</em> <code>s</code> <em>is <strong>valid</strong></em>.</p>

<p>The following rules define a <strong>valid</strong> string:</p>

<ul>
	<li>Any left parenthesis <code>&#39;(&#39;</code> must have a corresponding right parenthesis <code>&#39;)&#39;</code>.</li>
	<li>Any right parenthesis <code>&#39;)&#39;</code> must have a corresponding left parenthesis <code>&#39;(&#39;</code>.</li>
	<li>Left parenthesis <code>&#39;(&#39;</code> must go before the corresponding right parenthesis <code>&#39;)&#39;</code>.</li>
	<li><code>&#39;*&#39;</code> could be treated as a single right parenthesis <code>&#39;)&#39;</code> or a single left parenthesis <code>&#39;(&#39;</code> or an empty string <code>&quot;&quot;</code>.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre><strong>Input:</strong> s = "()"
<strong>Output:</strong> true
</pre><p><strong class="example">Example 2:</strong></p>
<pre><strong>Input:</strong> s = "(*)"
<strong>Output:</strong> true
</pre><p><strong class="example">Example 3:</strong></p>
<pre><strong>Input:</strong> s = "(*))"
<strong>Output:</strong> true
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= s.length &lt;= 100</code></li>
	<li><code>s[i]</code> is <code>&#39;(&#39;</code>, <code>&#39;)&#39;</code> or <code>&#39;*&#39;</code>.</li>
</ul>

class Solution {
private:
    bool rec(string s, int i, int x, vector<vector<int>> &dp){
        // base
        if(i==s.length() and x==0) return true;
        

        if(x<0) return false;
        else if(dp[i][x] != -1) return dp[i][x];

        bool ans = false;
        if(s[i] == '(')
            ans = rec(s,i+1,x+1,dp);
        else if(s[i] == ')')
            ans = rec(s,i+1, x-1,dp);
        else if(s[i] == '*')    
            ans = rec(s,i+1,x,dp) || rec(s,i+1,x+1,dp) || rec(s,i+1, x-1,dp);

        return dp[i][x] = ans;
    }
public:
    bool checkValidString(string s) {
        int n = s.length();
        vector<vector<int>> dp(n+1, vector<int>(n+1,-1));

        rec(s,0,0,dp);

        return dp[0][0];
        
        // return dp[n-1][0] || dp[n-1][1] || dp[n-1][3];
    }   
};
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