<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: 
    vector<vector<int>> dp;
    bool recur(int ind, int open,string& s){
        if(open<0){
            return false;
        }
        if(ind==s.size()){
            if(open==0)
                return true;
            return false;
        }
        if(dp[ind][open]!=-1){
            return dp[ind][open];
        }
        bool ans = false;
        if(s[ind]==')'){
            ans = ans | recur(ind+1, open-1, s);
        } else if(s[ind]=='('){
            ans = ans | recur(ind+1, open+1, s);
        } else {
            ans = ans | recur(ind+1, open+1, s);
            ans = ans | recur(ind+1,open-1, s);
            ans = ans | recur(ind+1, open, s);
        }
        return dp[ind][open] = ans;
    }
public:
    bool checkValidString(string s) {
        dp.resize(101,vector<int>(101,-1));
        return recur(0,0,s);
    }
};
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