C - Chinese Restaurant

Score : $300$ points

Problem Statement

Person $0$, Person $1$, $\ldots$, and Person $(N-1)$ are sitting around a turntable in their counterclockwise order, evenly spaced. Dish $p_i$ is in front of Person $i$ on the table.
You may perform the following operation $0$ or more times:

• Rotate the turntable by one $N$-th of a counterclockwise turn. As a result, the dish that was in front of Person $i$ right before the rotation is now in front of Person $(i+1) \bmod N$.

When you are finished, Person $i$ is happy if Dish $i$ is in front of Person $(i-1) \bmod N$, Person $i$, or Person $(i+1) \bmod N$.
Find the maximum possible number of happy people.

Constraints

• $3 \leq N \leq 2 \times 10^5$
• $0 \leq p_i \leq N-1$
• $p_i \neq p_j$ if $i \neq j$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$p_0$ $\ldots$ $p_{N-1}$

Output

Print the answer.

4
1 2 0 3

4

The figure below shows the table after one operation.

Here, there are four happy people:

• Person $0$ is happy because Dish $0$ is in front of Person $3\ (=(0-1) \bmod 4)$;
• Person $1$ is happy because Dish $1$ is in front of Person $1\ (=1)$;
• Person $2$ is happy because Dish $2$ is in front of Person $2\ (=2)$;
• Person $3$ is happy because Dish $3$ is in front of Person $0\ (=(3+1) \bmod 4)$.

There cannot be five or more happy people, so the answer is $4$.

3
0 1 2

3

10
3 9 6 1 7 2 8 0 5 4

5