Transition Game

AtCoder

IDabc296_e

Time2000ms

Memory256MB

Difficulty—

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: $N$ $A_1$ $A_2$ $\ldots$ $A_N$ [samples]

Samples

Input #1

3
2 2 3

Output #1

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 $2$.

Input #2

2
2 1

Output #2

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 $2$.

API Response (JSON)

{
  "problem": {
    "name": "Transition Game",
    "description": {
      "content": "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,\\",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc296_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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 $1\\leq A_i \\leq N$.\nTakahashi and Aoki will play $N$ rounds of a game. For each $i=1,2,\\...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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