<p>Find all valid combinations of <code>k</code> numbers that sum up to <code>n</code> such that the following conditions are true:</p>

<ul>
	<li>Only numbers <code>1</code> through <code>9</code> are used.</li>
	<li>Each number is used <strong>at most once</strong>.</li>
</ul>

<p>Return <em>a list of all possible valid combinations</em>. The list must not contain the same combination twice, and the combinations may be returned in any order.</p>

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

<pre>
<strong>Input:</strong> k = 3, n = 7
<strong>Output:</strong> [[1,2,4]]
<strong>Explanation:</strong>
1 + 2 + 4 = 7
There are no other valid combinations.</pre>

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

<pre>
<strong>Input:</strong> k = 3, n = 9
<strong>Output:</strong> [[1,2,6],[1,3,5],[2,3,4]]
<strong>Explanation:</strong>
1 + 2 + 6 = 9
1 + 3 + 5 = 9
2 + 3 + 4 = 9
There are no other valid combinations.
</pre>

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

<pre>
<strong>Input:</strong> k = 4, n = 1
<strong>Output:</strong> []
<strong>Explanation:</strong> There are no valid combinations.
Using 4 different numbers in the range [1,9], the smallest sum we can get is 1+2+3+4 = 10 and since 10 &gt; 1, there are no valid combination.
</pre>

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

<ul>
	<li><code>2 &lt;= k &lt;= 9</code></li>
	<li><code>1 &lt;= n &lt;= 60</code></li>
</ul>

class Solution {
    pair<bool, vector<vector<int>>> backtrack(int k, int start_digit, int n) {
        if (k == 0) {
            if (n == 0)
                return {true, {{}}};
            else
                return {false, {{}}};
        }

        vector<vector<int>> ans;
        for (int d = start_digit ; d <= 9 && d <= n ; d++) {
            auto [ok, suffixs] = backtrack(k-1, d+1, n-d);
            if (ok) {
                for (vector<int> &suffix : suffixs) {
                    vector<int> tmp = {d};
                    for (int s : suffix)
                        tmp.push_back(s);
                    ans.push_back(tmp);
                }
            }
        }

        return {ans.size() > 0, ans};
    }
public:
    vector<vector<int>> combinationSum3(int k, int n) {
        auto [ok, ans] = backtrack(k, 1, n);
        return ans;
    }
};
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