Reverse Brackets

AtCoder

IDarc194_d

Time2000ms

Memory256MB

Difficulty—

A string is defined to be a **valid parenthesis sequence** if and only if it satisfies one of the following conditions: * It is an empty string. * There exists a valid parenthesis sequence $A$ such that the string is obtained by concatenating `(`, $A$, and `)` in this order. * There exist non-empty valid parenthesis sequences $A$ and $B$ such that the string is obtained by concatenating $A$ and $B$ in this order. You are given a valid parenthesis sequence $S$ of length $N$. You can perform the following operation any number of times: * Choose a contiguous substring of $S$ that is a valid parenthesis sequence, and reverse it. Here, reversing the substring of $S$ from the $l$\-th character to the $r$\-th character means the following: * For every integer $i$ satisfying $l \leq i \leq r$, simultaneously replace $S_i$ with `)` if $S_{l+r-i}$ is `(`, and with `(` if $S_{l+r-i}$ is `)`.(Note that reversing here is different from the usual definition of reversing.) Find the number, modulo $998244353$, of distinct strings $S$ that you can have at the end of the process. ## Constraints * $1 \leq N \leq 5000$ * $|S| = N$ * $S$ is a valid parenthesis sequence. ## Input The input is given from Standard Input in the following format: $N$ $S$ [samples]

Samples

Input #1

6
(())()

Output #1

2

For example, you can transform $S$ into `()(())` by doing the following:

*   Choose the substring from the 1st to the 6th character of $S$. This is a valid parenthesis sequence. $S$ becomes `()(())`.

The only other string that can be formed is `(())()`. Thus, the answer is $2$.

Input #2

2
()

Output #2

API Response (JSON)

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  "problem": {
    "name": "Reverse Brackets",
    "description": {
      "content": "A string is defined to be a **valid parenthesis sequence** if and only if it satisfies one of the following conditions: *   It is an empty string. *   There exists a valid parenthesis sequence $A$ su",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc194_d"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "A string is defined to be a **valid parenthesis sequence** if and only if it satisfies one of the following conditions:\n\n*   It is an empty string.\n*   There exists a valid parenthesis sequence $A$ su...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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