Paw

AtCoder

IDarc132_e

Time2000ms

Memory256MB

Difficulty—

There are $n$ squares arranged in a row. Each square has a left- or right-pointing footprint or a hole. The initial state of each square is described by a string $s$ consisting of `.`, `<`, `>`. If the $i$\-th character of $s$ is `.`, the $i$\-th square from the left has a hole; if that character is `<`, that square has a left-pointing footprint; if that character is `>`, that square has a right-pointing footprint. Snuke, the cat, will repeat the following procedure until there is no more square with a hole. * Step $1$: Choose one square with a hole randomly with equal probability. * Step $2$: Fill the hole in the chosen square, stand there, and face to the left or right randomly with equal probability. * Step $3$: Keep walking in the direction Snuke is facing until he steps on a square with a hole or exits the row of squares. Here, the choices of squares and directions are independent of each other. When Snuke walks on a square (without a hole), that square gets a footprint in the direction he walks in. If the square already has a footprint, it gets erased and replaced by a new one. For example, in the situation `>.<><.><`, if Snuke chooses the $6$\-th square from the left and faces to the left, the $6$\-th, $5$\-th, $4$\-th, $3$\-rd squares get left-pointing footprints: `>.<<<<><`. Find the expected value of the number of left-pointing footprints when Snuke finishes the procedures, modulo $998244353$. ## Constraints * $1 \leq n \leq 10^5$ * $s$ is a string of length $n$ consisting of `.`, `<`, `>`. * $s$ contains at least one `.`. ## Input Input is given from Standard Input in the following format: $n$ $s$ [samples] ## Notes When the sought expected value is represented as an irreducible fraction $p/q$, there uniquely exists an integer $r$ such that $rq\equiv p ~(\text{mod } 998244353)$ and $0 \leq r \lt 998244353$ under the Constraints of this problem. This $r$ is the value to be found.

Samples

Input #1

5
><.<<

Output #1

3

In Step $1$, Snuke always chooses the only square with a hole.
If Snuke faces to the left in Step $2$, the squares become `<<<<<`, where $5$ squares have left-pointing footprints.
If Snuke faces to the right in Step $2$, the squares become `><>>>`, where $1$ square has a left-pointing footprint.
Therefore, the answer is $\frac{5+1}{2}=3$.

Input #2

20
>.>.<>.<<.<>.<..<>><

Output #2

848117770

API Response (JSON)

{
  "problem": {
    "name": "Paw",
    "description": {
      "content": "There are $n$ squares arranged in a row. Each square has a left- or right-pointing footprint or a hole. The initial state of each square is described by a string $s$ consisting of `.`, `<`, `>`. If th",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc132_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $n$ squares arranged in a row. Each square has a left- or right-pointing footprint or a hole. The initial state of each square is described by a string $s$ consisting of `.`, `<`, `>`. If th...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments