CF2018D Max Plus Min Plus Size

[EnV - The Dusty Dragon Tavern](https://soundcloud.com/envyofficial/env-the-dusty-dragon-tavern) ⠀ You are given an array $ a_1, a_2, \ldots, a_n $ of positive integers. You can color some elements of the array red, but there cannot be two adjacent red elements (i.e., for $ 1 \leq i \leq n-1 $ , at least one of $ a_i $ and $ a_{i+1} $ must not be red). Your score is the maximum value of a red element, plus the minimum value of a red element, plus the number of red elements. Find the maximum score you can get.

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10^4 $ ). The description of the test cases follows. The first line of each test case contains a single integer $ n $ ( $ 1 \le n \le 2 \cdot 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 \le a_i \le 10^9 $ ) — the given array. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case, output a single integer: the maximum possible score you can get after coloring some elements red according to the statement.

In the first test case, you can color the array as follows: $ [\color{red}{5}, 4, \color{red}{5}] $ . Your score is $ \max([5, 5]) + \min([5, 5]) + \text{size}([5, 5]) = 5+5+2 = 12 $ . This is the maximum score you can get. In the second test case, you can color the array as follows: $ [4, \color{red}{5}, 4] $ . Your score is $ \max([5]) + \min([5]) + \text{size}([5]) = 5+5+1 = 11 $ . This is the maximum score you can get. In the third test case, you can color the array as follows: $ [\color{red}{3}, 3, \color{red}{3}, 3, \color{red}{4}, 1, 2, \color{red}{3}, 5, \color{red}{4}] $ . Your score is $ \max([3, 3, 4, 3, 4]) + \min([3, 3, 4, 3, 4]) + \text{size}([3, 3, 4, 3, 4]) = 4+3+5 = 12 $ . This is the maximum score you can get.