1343D · Constant Palindrome Sum

Problem: You are given an array $a$ consisting of $n$ integers (it is guaranteed that $n$ is even, i.e. divisible by $2$). All $a_i$ does not exceed some integer $k$.

Your task is to replace the minimum number of elements (replacement is the following operation: choose some index $i$ from $1$ to $n$ and replace $a_i$ with some integer in range $[1; k]$) to satisfy the following conditions:

• after all replacements, all $a_i$ are positive integers not greater than $k$;
• for all $i$ from $1$ to $\frac{n}{2}$ the following equation is true: $a_i + a_{n - i + 1} = x$, where $x$ should be the same for all $\frac{n}{2}$ pairs of elements.

You have to answer $t$ independent test cases.

Input Format: The first line of the input contains one integer $t$ ($1 \le t \le 10^4$) — the number of test cases. Then $t$ test cases follow.

The first line of the test case contains two integers $n$ and $k$ ($2 \le n \le 2 \cdot 10^5, 1 \le k \le 2 \cdot 10^5$) — the length of $a$ and the maximum possible value of some $a_i$ correspondingly. It is guratanteed that $n$ is even (i.e. divisible by $2$). The second line of the test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le k$), where $a_i$ is the $i$-th element of $a$.

It is guaranteed that the sum of $n$ (as well as the sum of $k$) over all test cases does not exceed $2 \cdot 10^5$ ($\sum n \le 2 \cdot 10^5$, $\sum k \le 2 \cdot 10^5$).

Output Format: For each test case, print the answer — the minimum number of elements you have to replace in $a$ to satisfy the conditions from the problem statement.