<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 {

void print_st_s (string s, stack<char> st)
{
    cout << s <<endl;
    while (!st.empty())
    {
        cout << st.top();
        st.pop();
    }
}

    bool checkValidString(string &s, int pos, stack<char> &st)
    {
        if (pos == s.size())
        {
            return st.empty();
        }

        switch (s[pos])
            {
                case '(':
                    st.push (s[pos]);
                    return checkValidString (s, pos+1, st);                   

                case ')':
                    if (st.empty ())
                        return false;

                    st.pop ();
                    return checkValidString (s, pos+1, st);

                case '*':
                    stack<char> copy_st = st;
 
                    s[pos] = '(';
                    print_st_s (s, st);
                    if (checkValidString (s, pos, st))
                        return true;

                    st = copy_st;
                    s[pos] = ')';
                    if (checkValidString (s, pos, st))
                        return true;

                    st = copy_st;
                    return checkValidString (s, pos+1, st); 
            }

        assert (false);
        return false;
    }

public:
    bool checkValidString(string s) 
    {
        stack<char> st;
        return checkValidString (s, 0, st);
    }
};
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