CF1772D 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] $ .