Replace Triplets

AtCoder

IDarc187_e

Time2000ms

Memory256MB

Difficulty—

You are given a sequence $A = (A_1, \ldots, A_N)$ of length $N$. Here, $N$ is an integer not less than $3$. You can perform the following operation any number of times (zero or more). * Choose an integer $i$ satisfying $1 \leq i \leq N$ and $A_i = A_{i+1} = A_{i+2}$. Replace two of $A_i$, $A_{i+1}$, and $A_{i+2}$ with integers between $1$ and $N$, inclusive. Here, assume $A_{N+1} = A_1$ and $A_{N+2} = A_2$. Find the number, modulo $998244353$, of possible resulting sequences $A$ that are permutations of $(1, \ldots, N)$. ## Constraints * All input numbers are integers. * $3 \leq N \leq 5 \times 10^5$ * $1 \leq A_i \leq N$ ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $\ldots$ $A_N$ [samples]

Samples

Input #1

6
1 2 3 3 1 1

Output #1

360

For example, we can obtain the permutation $A = (1,2,4,3,5,6)$ through the following steps.

*   Perform the operation with $i=5$. Replace $A_5$ with $3$ and $A_6$ with $6$. Now, $A = (1,2,3,3,3,6)$.
*   Perform the operation with $i=3$. Replace $A_3$ with $4$ and $A_5$ with $5$. Now, $A = (1,2,4,3,5,6)$.

There are $360$ possible resulting sequences $A$ that are permutations of $(1, \ldots, 6)$.

Input #2

5
3 1 3 4 1

Output #2

0

There are no possible resulting sequences $A$ that are permutations of $(1, \ldots, 5)$.

Input #3

10
1 1 1 8 8 8 7 7 7 10

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Replace Triplets",
    "description": {
      "content": "You are given a sequence $A = (A_1, \\ldots, A_N)$ of length $N$. Here, $N$ is an integer not less than $3$. You can perform the following operation any number of times (zero or more). *   Choose an i",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc187_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a sequence $A = (A_1, \\ldots, A_N)$ of length $N$. Here, $N$ is an integer not less than $3$.\nYou can perform the following operation any number of times (zero or more).\n\n*   Choose an i...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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