<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 {
    map<int,map<size_t,int>> dp;

    bool search(string& s, int score, size_t i) {
        if (dp[score][i] == 2) {
            return true;
        }

        if (dp[score][i] == 1) {
            return false;
        }

        bool res;
        if (score < 0) {
            res = false;
        } else if (i == s.size()) {
            res = (score == 0);
        } else {
            if (s[i] == '(') {
                res = search(s, score + 1, i + 1);
            }

            if (s[i] == ')') {
                res = search(s, score - 1, i + 1);
            }

            if (s[i] == '*') {
                res = search(s, score + 1, i + 1)
                    || search(s, score - 1, i + 1)
                    || search(s, score, i + 1);
            }
        }

        dp[score][i] = res ? 2 : 1;

        return res;
    }

public:
    bool checkValidString(string s) {
        return search(s, 0, 0);
    }
};
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