D. Running Miles
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

There is a street with $$$n$$$ sights, with sight number $$$i$$$ being $$$i$$$ miles from the beginning of the street. Sight number $$$i$$$ has beauty $$$b_i$$$. You want to start your morning jog $$$l$$$ miles and end it $$$r$$$ miles from the beginning of the street. By the time you run, you will see sights you run by (including sights at $$$l$$$ and $$$r$$$ miles from the start). You are interested in the $$$3$$$ most beautiful sights along your jog, but every mile you run, you get more and more tired.

So choose $$$l$$$ and $$$r$$$, such that there are at least $$$3$$$ sights you run by, and the sum of beauties of the $$$3$$$ most beautiful sights minus the distance in miles you have to run is maximized. More formally, choose $$$l$$$ and $$$r$$$, such that $$$b_{i_1} + b_{i_2} + b_{i_3} - (r - l)$$$ is maximum possible, where $$$i_1, i_2, i_3$$$ are the indices of the three maximum elements in range $$$[l, r]$$$.

Input

The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^5$$$) — the number of test cases.

The first line of each test case contains a single integer $$$n$$$ ($$$3 \leq n \leq 10^5$$$).

The second line of each test case contains $$$n$$$ integers $$$b_i$$$ ($$$1 \leq b_i \leq 10^8$$$) — beauties of sights $$$i$$$ miles from the beginning of the street.

It's guaranteed that the sum of all $$$n$$$ does not exceed $$$10^5$$$.

Output

For each test case output a single integer equal to the maximum value $$$b_{i_1} + b_{i_2} + b_{i_3} - (r - l)$$$ for some running range $$$[l, r]$$$.

Example
Input
4
5
5 1 4 2 3
4
1 1 1 1
6
9 8 7 6 5 4
7
100000000 1 100000000 1 100000000 1 100000000
Output
8
1
22
299999996
Note

In the first example, we can choose $$$l$$$ and $$$r$$$ to be $$$1$$$ and $$$5$$$. So we visit all the sights and the three sights with the maximum beauty are the sights with indices $$$1$$$, $$$3$$$, and $$$5$$$ with beauties $$$5$$$, $$$4$$$, and $$$3$$$, respectively. So the total value is $$$5 + 4 + 3 - (5 - 1) = 8$$$.

In the second example, the range $$$[l, r]$$$ can be $$$[1, 3]$$$ or $$$[2, 4]$$$, the total value is $$$1 + 1 + 1 - (3 - 1) = 1$$$.