| [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. |