<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 checkValidString(string s) {
        map<pair<int,int>,bool> dp;
        return dfs(s,0,0,dp);
    }
private:
    bool dfs(string &s, int i, int openCount,map<pair<int,int>,bool> &dp){
        if (i == s.size()) {
            return openCount == 0;
        }
        if (openCount < 0) {
            return false;
        }
        if(dp.find({i,openCount}) != dp.end()){
            return dp[{i,openCount}];
        }
        bool ans = false;
        if(s[i] == '*'){
            ans = ans || dfs(s,i+1,openCount+1,dp);
            ans = ans || dfs(s,i+1,openCount-1,dp);
            ans = ans || dfs(s,i+1,openCount,dp);
        }
        else if(s[i] == ')'){
            ans = ans || dfs(s,i+1,openCount-1,dp);
        }
        else{
            ans = ans || dfs(s,i+1,openCount+1,dp);
        }
        dp[{i,openCount}] = ans;
        return ans;
    }
};
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