<p>Given an integer array <code>nums</code> and an integer <code>k</code>, return <code>true</code> if it is possible to divide this array into <code>k</code> non-empty subsets whose sums are all equal.</p>

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

<pre>
<strong>Input:</strong> nums = [4,3,2,3,5,2,1], k = 4
<strong>Output:</strong> true
<strong>Explanation:</strong> It is possible to divide it into 4 subsets (5), (1, 4), (2,3), (2,3) with equal sums.
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,2,3,4], k = 3
<strong>Output:</strong> false
</pre>

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

<ul>
	<li><code>1 &lt;= k &lt;= nums.length &lt;= 16</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>4</sup></code></li>
	<li>The frequency of each element is in the range <code>[1, 4]</code>.</li>
</ul>

static const bool Booster = [](){
    std::ios_base::sync_with_stdio(false);
    std::cout.tie(nullptr);
    std::cin.tie(nullptr);
    return true;
}();

class Solution {
public:
    bool backtrack(std::vector<int>& nums, std::vector<bool>& taken, int k, int numCompleted, int currentSum, int index, std::unordered_map<std::string, bool>& memo, int target) {
        std::string takenStr;
        for (bool t : taken) {
            takenStr += t ? '1' : '0';
        }

        if (numCompleted == k - 1) {
            return true;
        }

        if (currentSum > target) {
            return false;
        }

        if (memo.find(takenStr) != memo.end()) {
            return memo[takenStr];
        }

        if (currentSum == target) {
            memo[takenStr] = backtrack(nums, taken, k, numCompleted + 1, 0, 0, memo, target);
            return memo[takenStr];
        }

        for (int i = index; i < nums.size(); i++) {
            if (!taken[i]) {
                taken[i] = true;
                if (backtrack(nums, taken, k, numCompleted, currentSum + nums[i], i + 1, memo, target)) {
                    return true;
                }
                taken[i] = false;
            }

            if (std::all_of(taken.begin(), taken.end(), [](bool v) { return !v; })) {
                break;
            }
        }

        memo[takenStr] = false;
        return memo[takenStr];
    }

    bool canPartitionKSubsets(std::vector<int>& nums, int k) {
        int totalSum = std::accumulate(nums.begin(), nums.end(), 0);
        std::unordered_map<std::string, bool> memo;

        if (totalSum % k != 0) {
            return false;
        }

        int target = totalSum / k;
        std::vector<bool> taken(nums.size(), false);

        return backtrack(nums, taken, k, 0, 0, 0, memo, target);
    }
};

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