<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 {
public:
    int rec(int i, int sum, string s,vector<vector<int>>&dp){
        if(i==s.length()){
            if(sum) return 0;
            return 1;
        }
        if(sum<0) return 0;
        if(dp[i][sum]!=-1000)
            return dp[i][sum];
        if(s[i]=='('){
            return dp[i][sum] = rec(i+1,sum+1,s,dp);
        }
        if(s[i]==')'){
            return dp[i][sum]=rec(i+1,sum-1,s,dp);
        }
        if(s[i]=='*'){
            if(rec(i+1,sum+1,s,dp)==1||rec(i+1,sum-1,s,dp)==1||rec(i+1,sum,s,dp)==1)
                return dp[i][sum]=1;
            else
                return dp[i][sum]=0;
        }
        return 0;
    }
    bool checkValidString(string s) {
        
        int n=s.length();
        vector<vector<int>> dp(n+1,vector<int>(100,-1000));
        int ans=rec(0,0,s,dp);
        if(ans) return 1;
        return 0;
        
    }
};
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