Score : $300$ points

问题描述

设有$0$号、$1$号、$\ldots$及$(N-1)$号共$N$个人，他们按照逆时针顺序均匀地围坐在一个转盘周围。桌上碟子$p_i$位于第$i$号人面前。

你可以执行以下操作零次或多次：

• 将转盘逆时针旋转$\frac{1}{N}$圈。这样一来，在旋转前位于第$i$号人面前的碟子，现在会位于第$(i+1) \bmod N$号人面前。

操作结束后，如果碟子$i$位于第$(i-1) \bmod N$号人、第$i$号人或第$(i+1) \bmod N$号人面前，则认为第$i$号人是满意的。

求最多能使多少人感到满意。

什么是$a \bmod m$？对于整数$a$和正整数$m$，$a \bmod m$表示在$0$到$(m-1)$（包含两端）之间的一个整数$x$，满足$(a-x)$是$m$的倍数。（可以证明这样的$x$是唯一的。）

以上为通义千问 qwen-max 翻译，仅供参考。

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.

What is $a \bmod m$? For an integer $a$ and a positive integer $m$, $a \bmod m$ denotes the integer $x$ between $0$ and $(m-1)$ (inclusive) such that $(a-x)$ is a multiple of $m$. (It can be proved that such $x$ is unique.)

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.

Sample Input 1

4
1 2 0 3

Sample Output 1

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 $4$.

Sample Input 2

3
0 1 2

Sample Output 2

3

Sample Input 3

10
3 9 6 1 7 2 8 0 5 4

Sample Output 3

5

update @ 2024/3/10 11:18:07