<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 n ;
     unordered_map<string,bool>mp;
     
     bool solve(string &s,int ind,int cnt){

        if(cnt<0){
            return false;
        }

        if(ind==n){
            return cnt==0;
        }
        
        string key = to_string(ind)+ ',' + to_string(cnt);
        if(mp.find(key) != mp.end()){
               return mp[key];
        }

        bool result;

        if(s[ind]=='('){
          result  = solve(s,ind+1,cnt+1);
        }

        else if(s[ind]==')'){
            result = solve(s,ind+1,cnt-1);
        }

        else{
          result = solve(s,ind+1,cnt+1) || solve(s,ind+1,cnt-1) || solve(s,ind+1,cnt);
        }

         mp[key] = result;
         return result;

      }
      
    bool checkValidString(string s) {
       n = s.length();
       mp.clear();
      return solve(s,0,0);
    }
};
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