<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:

        """
        - minimum_x_to_plus_two
            - curr >= other, or other > curr
            - We want to maximize the abs(curr - other) to cover more 'x' to minimise +2s -> substitute k or 0.
            - if curr >= other -> max(k - other, curr)
            - if curr < other -> max(k - curr, other)
            - max( max(k - other, curr), max(k - curr, other) ) = max(k - other, curr, k - curr, other) 
        """

        n = len(nums)
        max_x = k + 1

        diffs = defaultdict(int)
        plus_two_at_x = defaultdict(int)
        for idx in range(len(nums) // 2):
            curr, other = nums[idx], nums[n-idx-1]
            diffs[abs(curr - other)] += 1
            minimum_x_to_plus_two = max(k - other, k - curr, curr, other) + 1 # if x == minimum_x_to_plus_two, need to plus 2.
            plus_two_at_x[minimum_x_to_plus_two] += 1
    
        res = n//2
        running_plus_twos = 0
        for x in range(max_x):
            running_plus_twos += plus_two_at_x[x] 
            skip = diffs[x] 
            plus_one = n//2 - running_plus_twos - skip
            res = min(res, running_plus_twos*2 + plus_one)
        return res
42100