<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

        visited = [False] * len(nums)
        cache = {}

        def dp(i, summ, div):
            # Cache key based on the current state of the function
            cache_key = (i, summ, div, tuple(visited))

            # Check if the result for this state is already cached
            if cache_key in cache:
                return cache[cache_key]

            if div == k - 1:  # if we created k subsets, return true
                return True

            if summ == target:  # If our current subset reaches the target, start a new subset
                result = dp(0, 0, div + 1)
                cache[cache_key] = result
                return result

            if i >= len(nums):  # Return false if we have no more numbers to use for our subset
                cache[cache_key] = False
                return False

            for l in range(i, len(nums)):
                if not visited[l] and summ + nums[l] <= target:  # if we did not use that number before and can add it to our subset
                    visited[l] = True
                    if dp(l + 1, summ + nums[l], div):  # Continue adding numbers to our subset
                        cache[cache_key] = True
                        return True
                    visited[l] = False

            cache[cache_key] = False
            return False

        return dp(0, 0, 0)

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