Score : $700$ points

Problem Statement

There are $N$ people, Person $1$ through Person $N$, standing on a number line. Person $i$ is now at coordinate $A_i$ on the number line. Here, $A_1 < A_2 < A_3 < \dots < A_N$ and all $A_i$ are even numbers.

We will now hold a party for $k$ seconds. During the party, each person can freely move on the number line at a speed of at most $1$ per second. (It is also allowed to move in the negative direction within the speed limit.) The people request that the following condition be satisfied for every integer $i$ such that $1 \le i \lt N$:

• there is at least one moment during the party (including the moment it ends) when Person $i$ and Person $i + 1$ are at the same coordinate.

Find the minimum $k$ for which there is a strategy that the people can take to satisfy all the conditions. We can prove that the answer is an integer under the Constraints of this problem.

Constraints

• $2 \le N \le 2 \times 10^5$
• $0 \le A_i \le 10^9$
• $A_1 < A_2 < A_3 < \dots < A_N$
• $A_i$ is an even number.

Input

Input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $A_3$ $\dots$ $A_N$

Output

Print the answer as an integer.

3
0 6 10

5

During the party lasting $5$ seconds, each person should move as follows:

• Person $1$: always moves in the positive direction at a speed of $1$.
• Person $2$: moves in the positive direction at a speed of $1$ during the first $2$ seconds, then in the negative direction at a speed of $1$ during the remaining $3$ seconds.
• Person $3$: always moves in the negative direction at a speed of $1$.

If they move in this way, Person $2$ and Person $3$ will be at the same coordinate exactly $2$ seconds after the beginning, and Person $1$ and Person $2$ will be at the same coordinate at the end of the party. Thus, they can satisfy all the conditions for $k = 5$. For smaller $k$, there is no strategy to satisfy all of them.

5
0 2 4 6 8

3

During the party lasting $3$ seconds, each person should, for example, move as follows:

• Person $1$: always moves in the positive direction at a speed of $1$.
• Person $2$: moves in the positive direction at a speed of $1$ during the first $2$ seconds, then in the negative direction at a speed of $1$ during the remaining $1$ second.
• Person $3$: does not move at all.
• Person $4$: moves in the negative direction at a speed of $1$ during the first $2$ seconds, then in the positive direction at a speed of $1$ during the remaining $1$ second.
• Person $5$: always moves in the negative direction at a speed of $1$.

10
0 2 4 6 8 92 94 96 98 100

44