CF1720B Interesting Sum

You are given an array $ a $ that contains $ n $ integers. You can choose any proper subsegment $ a_l, a_{l + 1}, \ldots, a_r $ of this array, meaning you can choose any two integers $ 1 \le l \le r \le n $ , where $ r - l + 1 < n $ . We define the beauty of a given subsegment as the value of the following expression: $$\max(a_{1}, a_{2}, \ldots, a_{l-1}, a_{r+1}, a_{r+2}, \ldots, a_{n}) - \min(a_{1}, a_{2}, \ldots, a_{l-1}, a_{r+1}, a_{r+2}, \ldots, a_{n}) + \max(a_{l}, \ldots, a_{r}) - \min(a_{l}, \ldots, a_{r}). $$ Please find the maximum beauty among all proper subsegments.

The first line contains one integer $ t $ ( $ 1 \leq t \leq 1000 $ ) — the number of test cases. Then follow the descriptions of each test case. The first line of each test case contains a single integer $ n $ $ (4 \leq n \leq 10^5) $ — the length of the array. The second line of each test case contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 1 \leq a_{i} \leq 10^9 $ ) — the elements of the given array. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 10^5 $ .

For each testcase print a single integer — the maximum beauty of a proper subsegment.

In the first test case, the optimal segment is $ l = 7 $ , $ r = 8 $ . The beauty of this segment equals to $ (6 - 1) + (5 - 1) = 9 $ . In the second test case, the optimal segment is $ l = 2 $ , $ r = 4 $ . The beauty of this segment equals $ (100 - 2) + (200 - 1) = 297 $ .