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

<p>You want to choose a subsequence of <code>nums</code> such that the sum of its elements is the closest possible to <code>goal</code>. That is, if the sum of the subsequence&#39;s elements is <code>sum</code>, then you want to <strong>minimize the absolute difference</strong> <code>abs(sum - goal)</code>.</p>

<p>Return <em>the <strong>minimum</strong> possible value of</em> <code>abs(sum - goal)</code>.</p>

<p>Note that a subsequence of an array is an array formed by removing some elements <strong>(possibly all or none)</strong> of the original array.</p>

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

<pre>
<strong>Input:</strong> nums = [5,-7,3,5], goal = 6
<strong>Output:</strong> 0
<strong>Explanation:</strong> Choose the whole array as a subsequence, with a sum of 6.
This is equal to the goal, so the absolute difference is 0.
</pre>

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

<pre>
<strong>Input:</strong> nums = [7,-9,15,-2], goal = -5
<strong>Output:</strong> 1
<strong>Explanation:</strong> Choose the subsequence [7,-9,-2], with a sum of -4.
The absolute difference is abs(-4 - (-5)) = abs(1) = 1, which is the minimum.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,3], goal = -7
<strong>Output:</strong> 7
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 40</code></li>
	<li><code>-10<sup>7</sup> &lt;= nums[i] &lt;= 10<sup>7</sup></code></li>
	<li><code>-10<sup>9</sup> &lt;= goal &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def minAbsDifference(self, nums: List[int], goal: int) -> int:
        # Helper function to generate all possible sums of subsequences of an array
        def generate_sums(arr):
            sums = set()
            n = len(arr)
            # Iterate over all possible subsequences using combinations
            for i in range(n + 1):
                for comb in combinations(arr, i):
                    sums.add(sum(comb))
            return list(sums)

        # Step 1: Split the array into two halves
        n = len(nums)
        left_part = nums[:n // 2]
        right_part = nums[n // 2:]
        
        # Step 2: Generate all possible sums for each half
        left_sums = generate_sums(left_part)
        right_sums = generate_sums(right_part)
        
        # Step 3: Sort the right_sums to perform binary search
        right_sums.sort()
        
        # Step 4: Initialize the minimum difference
        min_diff = abs(goal)  # Start with the difference between goal and 0 (choosing no elements)

        # Step 5: For each sum in left_sums, find the closest sum in right_sums to minimize abs(sum - goal)
        for left_sum in left_sums:
            target = goal - left_sum
            # Binary search for the closest sum in right_sums
            pos = bisect_left(right_sums, target)
            
            # Check the exact match or the closest values around the found position
            if pos < len(right_sums):
                min_diff = min(min_diff, abs(left_sum + right_sums[pos] - goal))
            if pos > 0:
                min_diff = min(min_diff, abs(left_sum + right_sums[pos - 1] - goal))

        return min_diff
154500