<p>Given a string <code>s</code> that contains parentheses and letters, remove the minimum number of invalid parentheses to make the input string valid.</p>

<p>Return <em>a list of <strong>unique strings</strong> that are valid with the minimum number of removals</em>. You may return the answer in <strong>any order</strong>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;()())()&quot;
<strong>Output:</strong> [&quot;(())()&quot;,&quot;()()()&quot;]
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;(a)())()&quot;
<strong>Output:</strong> [&quot;(a())()&quot;,&quot;(a)()()&quot;]
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;)(&quot;
<strong>Output:</strong> [&quot;&quot;]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= s.length &lt;= 25</code></li>
	<li><code>s</code> consists of lowercase English letters and parentheses <code>&#39;(&#39;</code> and <code>&#39;)&#39;</code>.</li>
	<li>There will be at most <code>20</code> parentheses in <code>s</code>.</li>
</ul>

class Solution {
public:

    void dfs (string &s, int index, int left_removed, int right_removed, int pair, string path, unordered_set<string> &result)
    {
        if (index >= s.size ()){
            if (left_removed == 0 && right_removed == 0 && pair == 0)
                result.insert (path);
        }
        else{
            if (s[index] == '('){
                if (left_removed > 0)
                    dfs (s, index + 1, left_removed - 1, right_removed, pair, path, result);
                dfs (s, index + 1, left_removed, right_removed, pair + 1, path + s[index], result);
            }
            else if (s[index] == ')'){
                if (right_removed > 0)
                    dfs (s, index + 1, left_removed, right_removed - 1, pair, path, result);
                if (pair > 0)
                    dfs (s, index + 1, left_removed, right_removed, pair - 1, path + s[index], result);
            }
            else
                dfs (s, index + 1, left_removed, right_removed, pair, path + s[index], result);
        }
    }
    vector<string> removeInvalidParentheses(string s) {
        unordered_set<string> result;
        int left_removed = 0;
        int right_removed = 0;
        for (auto c : s){
            if (c == '('){
                left_removed++;
            }
            else if (c == ')'){
                if (left_removed != 0)
                    left_removed--;
                else
                    right_removed++;
            }
        }
        dfs (s, 0, left_removed, right_removed, 0, "", result);
        return vector<string> (result.begin (), result.end ());
    }
};
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