A15620. Absolute Sorting

You are given an array $a$ consisting of $n$ integers. The array is sorted if $a_1 \le a_2 \le \dots \le a_n$ .

You want to make the array $a$ sorted by applying the following operation exactly once:

- choose an integer $x$ , then for every $i \in [1, n]$ , replace $a_i$ by $|a_i - x|$ .

Find any value of $x$ that will make the array sorted, or report that there is no such value.

The first line contains one integer $t$ ( $1 \le t \le 2 \cdot 10^4$ ) — the number of test cases.

Each test case consists of two lines. The first line contains one integer $n$ ( $2 \le n \le 2 \cdot 10^5$ ). The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ( $1 \le a_i \le 10^8$ ).

Additional constraint on the input: the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case, print any integer $x$ ( $0 \le x \le 10^9$ ) that makes the array sorted. It can be shown that if such an integer $x$ exists, there is at least one such integer between $0$ and $10^9$ .

If there is no such integer, then print $-1$ . If there are multiple suitable values of $x$ , print any of them.

In the first test case, after using $x = 4$ , the array becomes $[1, 1, 1, 1, 1]$ .

In the third test case, after using $x = 0$ , the array becomes $[1, 2, 3, 4, 5, 6, 7, 8]$ .

In the fourth test case, after using $x = 42$ , the array becomes $[32, 37, 38, 39, 40, 41]$ .