<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:

    bool solve(string s, int index, int count, vector<vector<int>> &dp){

        // base case
        if(count < 0){
            return false;
        }
        if(index >= s.length()){
            return count == 0;
        }

        if(dp[index][count] != -1){
            if(dp[index][count] == 1){
                return true;
            }
            else if(dp[index][count] == 0){
                return false;
            }
        }

        bool ans;

        if(s[index] == '('){

            ans = solve(s, index+1, count+1, dp);

            if(ans){
                dp[index][count] = 1;
            }
            else{
                dp[index][count] = 0;
            }
            return ans;
        }

        else if(s[index] == ')'){

            ans = solve(s, index+1, count-1, dp);

            if(ans){
                dp[index][count] = 1;
            }
            else{
                dp[index][count] = 0;
            }
            return ans;
        }
        
        ans = solve(s, index+1, count, dp) || solve(s, index+1, count+1, dp) || solve(s, index+1, count-1, dp);

        if(ans){
            dp[index][count] = 1;
        }
        else{
            dp[index][count] = 0;
        }
        return ans;
    }

    bool checkValidString(string s) {
        
        int count = 0;
        vector<vector<int>> dp(s.length(), vector<int> (s.length()+1, -1));
        return solve(s, 0, count, dp);
    }
};
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