<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(const string s)
    {
        string t;
        unordered_map<State, bool, StateHasher> memo;
        return isValid(0, s, t, memo);
    }

private:
    struct State
    {
        int i = 0;
        string s;

        bool operator==(const State &other) const { return i == other.i && s == other.s; }
    };

    struct StateHasher
    {
        size_t operator()(const State &state) const
        {
            const auto [i, s] = state;

            const size_t iHash = hash<int>{}(i);
            const size_t sHash = hash<string>{}(s);

            return (iHash << 1) ^ sHash;
        }
    };

    bool isValid(const int index, const string &parenthesis, string &s, unordered_map<State, bool, StateHasher> &memo)
    {
        const auto n = static_cast<int>(parenthesis.size());
        if (index == n)
        {
            return s.empty();
        }

        const State state { index, s };

        if (memo.count(state))
        {
            return memo[state];
        }

        const char current = parenthesis[index];

        if (current == '*')
        {
            if (isValid(index + 1, parenthesis, s, memo))
            {
                return memo[state] = true;
            }
        }

        if (current == '(' || current == '*')
        {
            s.push_back('(');

            if (isValid(index + 1, parenthesis, s, memo))
            {
                return memo[state] = true;
            }

            s.pop_back();
        }

        if (current == ')' || current == '*')
        {
            if (!s.empty() && s.back() == '(')
            {
                s.pop_back();

                if (isValid(index + 1, parenthesis, s, memo))
                {
                    return memo[state] = true;
                }

                s.push_back('(');
            }
        }

        return memo[state] = false;
    }
};
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