<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>

class Solution:
    def canPartitionKSubsets(self, nums: List[int], k: int) -> bool:
        total_sum = sum(nums)
        if total_sum % k != 0:
            return False
        
        target = total_sum // k
        nums.sort(reverse=True)
        if nums[0] > target:
            return False

        used = [0] * len(nums)
        
        # Memoization to store visited states
        memo = {}

        def backtrack(start, curr_sum, k):
            if k == 1:
                return True  # If only one group is left, the rest must fit
            if curr_sum == target:
                return backtrack(0, 0, k - 1)  # Found a valid subset, look for the next subset
            state = tuple(used)
            if state in memo:
                return memo[state]

            for i in range(start, len(nums)):
                if not used[i] and curr_sum + nums[i] <= target:
                    used[i] = 1
                    if backtrack(i + 1, curr_sum + nums[i], k):
                        memo[state] = True  # Memoize the success state
                        return True
                    used[i] = 0  # Backtrack

            memo[state] = False  # Memoize the failure state
            return False

        return backtrack(0, 0, k)

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