<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:
// similar to wildcard matching
bool check(int ind,int len,int c,string &s,vector<vector<int>>&dp)
{
    if(c<0)
    {
        return false;
    }
    // writing the base case
    if(ind==len)
    {
if(c==0)
{
    return true;
}
if(c!=0)
{
    return false;
}
    }
    bool ans;
    if(dp[ind][c]!=-1)
    {
        return dp[ind][c];
    }
    if(s[ind]=='(')
    {
        ans=check(ind+1,len,c+1,s,dp);
    }
    else if(s[ind]==')')
    {
       ans= check(ind+1,len,c-1,s,dp);
    }
    else
    {
        // * can mean nothing 
        // can mean '('
        // can mean ')'
        // if((check(ind+1,len,c,s,dp)|| check(ind+1,len,c+1,s,dp)|| check(ind+1,len,c-1,s,dp))==true)
        // {
        //     return true;
        // }
        ans =check(ind+1,len,c,s,dp)|| check(ind+1,len,c+1,s,dp)|| check(ind+1,len,c-1,s,dp);
    }
    return dp[ind][c]=ans;
}
    bool checkValidString(string s) {
        int len=s.length();
        int c=0;
        vector<vector<int>>dp(len,vector<int>(1000,-1));
        return check(0,len,c,s,dp);
        
    }
};
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