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E - Transition Game

Score : $500$ points

Problem Statement

You are given a sequence of $N$ numbers: $A=(A_1,A_2,\ldots,A_N)$. Here, each $A_i$ $(1\leq i\leq N)$ satisfies $1\leq A_i \leq N$.

Takahashi and Aoki will play $N$ rounds of a game. For each $i=1,2,\ldots,N$, the $i$-th game will be played as follows.

1. Aoki specifies a positive integer $K_i$.
2. After knowing $K_i$ Aoki has specified, Takahashi chooses an integer $S_i$ between $1$ and $N$, inclusive, and writes it on a blackboard.

3. Repeat the following $K_i$ times.

   • Replace the integer $x$ written on the blackboard with $A_x$.

If $i$ is written on the blackboard after the $K_i$ iterations, Takahashi wins the $i$-th round; otherwise, Aoki wins.
Here, $K_i$ and $S_i$ can be chosen independently for each $i=1,2,\ldots,N$.

Find the number of rounds Takahashi wins if both players play optimally to win.

Constraints

• $1\leq N\leq 2\times 10^5$
• $1\leq A_i\leq N$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\ldots$ $A_N$

Output

Find the number of rounds Takahashi wins if both players play optimally to win.

3
2 2 3

2

In the first round, if Aoki specifies $K_1=2$, Takahashi cannot win whichever option he chooses for $S_1$: $1$, $2$, or $3$.

For example, if Takahashi writes $S_1=1$ on the initial blackboard, the two operations will change this number as follows: $1\to 2(=A_1)$, $2\to 2(=A_2)$. The final number on the blackboard will be $2(\neq 1)$, so Aoki wins.

On the other hand, in the second and third rounds, Takahashi can win by writing $2$ and $3$, respectively, on the initial blackboard, whatever value Aoki specifies as $K_i$.

Therefore, if both players play optimally to win, Takashi wins two rounds: the second and the third. Thus, you should print $2$.

2
2 1

2

In the first round, Takahashi can win by writing $2$ on the initial blackboard if $K_1$ specified by Aoki is odd, and $1$ if it is even.

Similarly, there is a way for Takahashi to win the second round. Thus, Takahashi can win both rounds: the answer is $2$.