Duck Button Game

Background

Qiqi and her friends are playing a game on a one-dimensional grid consisting of $n$ cells numbered from $1$ to $n$ left to right.

Problem Description

Each cell has a button. If at any moment there are at least $a[i]$ ducks on cell $i$, the button on that cell is permanently pressed. Once pressed, it stays pressed even if the ducks leave. To win the game, all $n$ buttons must be pressed.

Initially, there are $d$ ducks on cell $1$. In each move, a single duck may move one cell to the left or to the right.

Determine the minimum total number of moves required to win the game. It is guaranteed that there exists a way to win the game.

Input Format

The input is given from standard input in the following format.

$n$ $d$ $a[1]$ $a[2]\dots a[n]$

Output Format

Print a single integer, the minimum total number of moves to win the game.

Sample

2 199
175 42

42

5 3
1 1 0 1 0

3

5 7
2 2 2 2 2

8

7 5
1 3 3 4 5 5 5

30

8 9
7 6 6 6 3 3 3 1

28

8 5
2 3 5 1 4 2 1 0

21

4 1000000000
1 1 1 999999999

2999999997

Constraints

$n$ up to $200,000$.

$d$ up to $10^{9}$.

$0 \le a[i] \le d$ for all $1 \le i \le n$.

It is guaranteed that there exists a way to win the game.