Bessie is experimenting with a powerful hoof implant that has the ability to create massive shock waves. She has $N$ ($2 \leq N \leq 10^5$) tiles lined up in front of her, which require powers of at least $p_0,p_1,\dots,p_{N-1}$ to break, respectively ($0 \leq p_i \leq 10^{18}$).

Bessie can apply power by punching a specific tile but due to the strange nature of her implant, it will not apply any power to the tile she punches. Instead, if she chooses to punch tile $x$ once, where $x$ is an integer in $[0,N-1]$, it applies $|i-x|$ power to tile $i$ for all integers $i$ in the range $[0,N-1]$. This power is also cumulative, so applying $2$ power twice to a tile will apply a total of $4$ power to the tile.

Please determine the fewest number of punches required to break all the tiles.

SAMPLE INPUT:

6
5
0 2 4 5 8
5
6 5 4 5 6
5
1 1 1 1 1
5
12 10 8 6 4
7
6 1 2 3 5 8 13
2
1000000000000000000 1000000000000000000

SAMPLE OUTPUT:

2
3
2
4
4
2000000000000000000

For the first test, the only way for Bessie to break all the tiles with two punches is to punch $0$ twice, applying total powers of $[0,2,4,6,8]$ respectively.

For the second test, one way for Bessie to break all the tiles with three punches is to punch $0$, $2$, and $4$ one time each, applying total powers of $[6,5,4,5,6]$ respectively.

For the third test, one way for Bessie to break all the tiles with two punches is to punch $0$ and $1$ one time each, applying total powers of $[1,1,3,5,7]$ respectively.

Problem credits: Suhas Nagar