<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:
    bool isValid(string x) {
        stack<char> st;
        for (char c : x) {
            if (c == '(') {
                st.push(c);
            } else if (c == ')') {
                if (!st.empty() && st.top() == '(') {
                    st.pop();
                } else {
                    st.push(c);
                }
            } else if (c >= 'a' && c <= 'z') {
                continue; 
            } else {
                st.push(c); 
            }
        }
        return st.empty();
    }

    void dfs(int i, int x, int y, string& s, string& ans, set<string>& all) {
        if (x < 0 || y < 0) return;
        if (i == s.length()) {
            if (x == 0 && y == 0 && isValid(ans)) {
                all.insert(ans);
            }
            return;
        }

        if (s[i] == '(') {
            dfs(i + 1, x - 1, y, s, ans, all);
            ans.push_back(s[i]);
            dfs(i + 1, x, y, s, ans, all);
            ans.pop_back();
        } 
        else if (s[i] == ')') {
            dfs(i + 1, x, y - 1, s, ans, all);
            ans.push_back(s[i]);
            dfs(i + 1, x, y, s, ans, all);
            ans.pop_back();
        } 
        else {
            ans.push_back(s[i]);
            dfs(i + 1, x, y, s, ans, all);
            ans.pop_back();
        }
    }

    vector<string> removeInvalidParentheses(string s) {
        stack<char> st;
        for (char c : s) {
            if (c == '(') {
                st.push(c);
            } else if (c == ')') {
                if (!st.empty() && st.top() == '(') {
                    st.pop();
                } else {
                    st.push(c);
                }
            }
        }

        int x = 0, y = 0;
        while (!st.empty()) {
            if (st.top() == '(') {
                x++;
            } else {
                y++;
            }
            st.pop();
        }

        set<string> all;
        string ans = "";
        dfs(0, x, y, s, ans, all);

        vector<string> result(all.begin(), all.end());
        return result;
    }
};
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