<p>You are given an integer array <code>nums</code> of size <code>n</code> where <code>n</code> is <strong>even</strong>, and an integer <code>k</code>.</p>

<p>You can perform some changes on the array, where in one change you can replace <strong>any</strong> element in the array with <strong>any</strong> integer in the range from <code>0</code> to <code>k</code>.</p>

<p>You need to perform some changes (possibly none) such that the final array satisfies the following condition:</p>

<ul>
	<li>There exists an integer <code>X</code> such that <code>abs(a[i] - a[n - i - 1]) = X</code> for all <code>(0 &lt;= i &lt; n)</code>.</li>
</ul>

<p>Return the <strong>minimum</strong> number of changes required to satisfy the above condition.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [1,0,1,2,4,3], k = 4</span></p>

<p><strong>Output:</strong> <span class="example-io">2</span></p>

<p><strong>Explanation:</strong><br />
We can perform the following changes:</p>

<ul>
	<li>Replace <code>nums[1]</code> by 2. The resulting array is <code>nums = [1,<u><strong>2</strong></u>,1,2,4,3]</code>.</li>
	<li>Replace <code>nums[3]</code> by 3. The resulting array is <code>nums = [1,2,1,<u><strong>3</strong></u>,4,3]</code>.</li>
</ul>

<p>The integer <code>X</code> will be 2.</p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [0,1,2,3,3,6,5,4], k = 6</span></p>

<p><strong>Output:</strong> <span class="example-io">2</span></p>

<p><strong>Explanation:</strong><br />
We can perform the following operations:</p>

<ul>
	<li>Replace <code>nums[3]</code> by 0. The resulting array is <code>nums = [0,1,2,<u><strong>0</strong></u>,3,6,5,4]</code>.</li>
	<li>Replace <code>nums[4]</code> by 4. The resulting array is <code>nums = [0,1,2,0,<strong><u>4</u></strong>,6,5,4]</code>.</li>
</ul>

<p>The integer <code>X</code> will be 4.</p>
</div>

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

<ul>
	<li><code>2 &lt;= n == nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>n</code> is even.</li>
	<li><code>0 &lt;= nums[i] &lt;= k &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution:
    def minChanges(self, nums: List[int], k: int) -> int:

        REV = lambda x: tuple(reversed(tuple(x)))

        LEN = len(nums)
        HALF = LEN//2
        firstHalf = tuple(nums[:HALF])
        secondRev = REV(nums[HALF:])
        # print(f'{firstHalf=}')
        # print(f'{secondRev=}')

        pairs = tuple(zip(firstHalf, secondRev))
        diffs = tuple([abs(A - B) for (A, B) in pairs])
        # print(f'{pairs=}')
        # print(f'{diffs=}')

        diffCounts = Counter(diffs)
        # print(f'{diffCounts=}')

        def movesRequiredToReachX(X: int, pair: Tuple[int,int]) -> int:
            # nonlocal k
            (A, B) = pair
            diff = abs(A - B)
            if diff == X:
                # print(f'  checkX({X},{pair}): {0}')
                return 0
            needed = [
                N
                for AB in pair
                for N in (AB - X, AB + X)
            ]
            allowed = [
                N
                for N in needed
                if 0 <= N <= k
            ]
            if allowed:
                # print(f'  checkX({X},{pair}): {1}, {allowed=}')
                return 1
            else:
                # print(f'  checkX({X},{pair}): {2}, invalid {needed=}')
                return 2

        def movesRequiredToReachXEverwhere(X: int) -> int:
            # nonlocal pairs
            return sum([
                movesRequiredToReachX(X, pair)
                for pair in pairs
            ])
        
        movesRequired = []
        for X, count in diffCounts.most_common(3):
            print(f'Trying {X=} ({count=}):')
            moves = movesRequiredToReachXEverwhere(X)
            print(f'  {moves=}')
            movesRequired.append(moves)
        
        print(f'{movesRequired=}')
        return min(movesRequired)

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