<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:
    bool check(string &s, int i, int op, unordered_map<long, bool> &umap){
        if(i >= s.size() && op == 0) return true;
        if(i >= s.size() || op < 0) return false;

        long key = ((long)i)<< 32 | long(op);
        if(umap.find(key) != umap.end()){
            return umap[key];
        }
        if(s[i] == '(') {
            umap[key] = check(s, i+1, op+1, umap);
        }
        else if(s[i] == ')'){
            umap[key] =  check(s, i+1, op-1, umap);
        }else {
            bool open =  check(s, i+1, op+1, umap);

            bool close =  check(s, i+1, op-1, umap);

            bool ignore = check(s, i+1, op, umap);

            umap[key] =  open || close || ignore;
        }
        return umap[key];
    }
public:
    bool checkValidString(string s) {
        int n = s.size();
        unordered_map<long, bool> umap;
        return check(s, 0, 0, umap);
    }
};

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