<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:
    unordered_map<int, set<string>> allPossibilities;
    int minRemove = INT_MAX;

    void search(int idx, string&s, int balance, int removes, string wip) {
        if (idx == s.size()-1){
            if (s[idx] == '('){
                if (balance == 0){
                    removes += 1;
                    if (removes <= minRemove) {
                        minRemove = removes;
                        allPossibilities[minRemove].insert(wip);
                    }
                }
            }else if (s[idx] == ')'){
                if (balance == 0){
                    removes += 1;
                    if (removes <= minRemove) {
                        minRemove = removes;
                        allPossibilities[minRemove].insert(wip);
                    }
                }else if (balance == 1){
                    wip += ')';
                    if (removes <= minRemove) {
                        minRemove = removes;
                        allPossibilities[minRemove].insert(wip);
                    }
                }
            }else{
                if (balance == 0){
                    wip += s[idx];
                    if (removes <= minRemove) {
                        minRemove = removes;
                        allPossibilities[minRemove].insert(wip);
                    }
                }
            }
        }else{
            if (s[idx] == '('){
                search(idx+1, s, balance+1, removes, wip+'(');
                search(idx+1, s, balance, removes+1, wip);
            }else if (s[idx] == ')'){
                if (balance > 0){
                    search(idx+1, s, balance-1, removes, wip+')');
                }
                search(idx+1, s, balance, removes+1, wip);
            }else{
                search(idx+1, s, balance, removes, wip+s[idx]);
            }
        }
    }

    vector<string> removeInvalidParentheses(string s) {
        // 10:46 PM - 11:16 PM
        vector<string> ans;
        
        int idx = 0;
        int balance = 0;
        int removes = 0;
        string wip = "";
        search(idx, s, balance, removes, wip);

        ans = vector<string>(allPossibilities[minRemove].begin(), allPossibilities[minRemove].end());

        return ans;
    }
};
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