<p>You are given an integer array <code>nums</code>.</p>

<p>You should move each element of <code>nums</code> into one of the two arrays <code>A</code> and <code>B</code> such that <code>A</code> and <code>B</code> are non-empty, and <code>average(A) == average(B)</code>.</p>

<p>Return <code>true</code> if it is possible to achieve that and <code>false</code> otherwise.</p>

<p><strong>Note</strong> that for an array <code>arr</code>, <code>average(arr)</code> is the sum of all the elements of <code>arr</code> over the length of <code>arr</code>.</p>

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

<pre>
<strong>Input:</strong> nums = [1,2,3,4,5,6,7,8]
<strong>Output:</strong> true
<strong>Explanation:</strong> We can split the array into [1,4,5,8] and [2,3,6,7], and both of them have an average of 4.5.
</pre>

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

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

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 30</code></li>
	<li><code>0 &lt;= nums[i] &lt;= 10<sup>4</sup></code></li>
</ul>

from functools import lru_cache

class Solution:
    def splitArraySameAverage(self, nums: List[int]) -> bool:
        n = len(nums)
        total_sum = sum(nums)
        
        if n == 1:
            return False
        
        nums.sort()
        
        possible = False
        for size in range(1, n // 2 + 1):
            if total_sum * size % n == 0:
                possible = True
                break
        if not possible:
            return False
        
        @lru_cache(None)
        def can_partition(size, target, index):
            if size == 0:
                return target == 0
            if target < 0 or size + index > n:
                return False
            return can_partition(size - 1, target - nums[index], index + 1) or can_partition(size, target, index + 1)
        
        for size in range(1, n // 2 + 1):
            if total_sum * size % n == 0:
                target = total_sum * size // n
                if can_partition(size, target, 0):
                    return True
        
        return False

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