<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 {
    bool check(string s, int i, int cnt, vector<vector<int>>& dp) {
        if (cnt < 0)
            return false;
        if (i == s.size())
            return cnt == 0;
        if (dp[i][cnt] != -1)
            return dp[i][cnt];

        if (s[i] == '(')
            return dp[i][cnt] = check(s, i + 1, cnt + 1, dp);
        else if (s[i] == ')')
            return dp[i][cnt] = check(s, i + 1, cnt - 1, dp);

        return dp[i][cnt] = check(s, i + 1, cnt + 1, dp) || 
                            check(s, i + 1, cnt - 1, dp) || 
                            check(s, i + 1, cnt, dp);
    }

public:
    bool checkValidString(string s) {
        int n = s.size();
        vector<vector<int>> dp(n, vector<int>(n * 2, -1)); 
        return check(s, 0, 0, dp);
    }
};

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