<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:
        def _td(nums, k):
            total = sum(nums)
            if total % k != 0:
                return False
            target = total // k 

            @cache 
            def _dfs(cur_sum, rem, rem_k):

                # stop condition
                if rem_k == 0:
                    return True
                
                # logic 
                ans = False
                for i in range(len(rem)):
                    new_sum = cur_sum + rem[i]
                    if new_sum == target:
                        ans |= _dfs(0, rem[:i] + rem[i+1:], rem_k-1)
                    elif new_sum < target:
                        ans |= _dfs(new_sum, rem[:i] + rem[i+1:], rem_k)
                    if ans:
                        return True
                
                return ans

            return _dfs(0, tuple(nums), k)

        return _td(nums, k)


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