Sugoroku2

AtCoder

IDabc189_f

Time2000ms

Memory256MB

Difficulty—

Takahashi is playing sugoroku. The board has $N+1$ squares numbered $0$ to $N$. Takahashi starts at Square $0$ and head to Square $N$. In this sugoroku, we use a wheel showing numbers from $1$ through $M$ with equal probability. In each turn, Takahashi spins the wheel and advances by the number shown by the wheel. When it makes him reach Square $N$ or go past it, he wins. Some of the squares send him to Square $0$ when he stops on them. There are $K$ such squares: Square $A_1, \ldots, A_K$. Find the expected value of the number of times Takahashi spins the wheel before he wins. If it is impossible to win, print `-1` instead. ## Constraints * All values in input are integers. * $1 \leq N \leq 10^5$ * $1 \leq M \leq 10^5$ * $0 \leq K \leq 10$ * $0 < A_1 < \ldots < A_K < N$ ## Input Input is given from Standard Input in the following format: $N$ $M$ $K$ $A_1$ $\ldots$ $A_K$ [samples]

Samples

Input #1

2 2 0

Output #1

1.5000

If the wheel shows $1$ in the first spin, he will need two spins to win; if the wheel shows $2$ in the first spin, he will need one spin to win. Thus, the expected number of spins is $1.5$.

Input #2

2 2 1
1

Output #2

2.0000

If the wheel shows $1$, he advances to Square $1$, but it sends him back to Square $0$.  
Thus, he keeps spinning the wheel until he gets $2$ and wins.  
The probability that he gets $2$ for the first time in the $i$\-th spin is $\frac{1}{2^i}$, so the expected number of spins is $\sum_{i = 1}^{\infty} (i \times \frac{1}{2^i}) = 2$.

Input #3

100 6 10
11 12 13 14 15 16 17 18 19 20

Output #3

\-1

Input #4

100000 2 2
2997 92458

Output #4

201932.2222

API Response (JSON)

{
  "problem": {
    "name": "Sugoroku2",
    "description": {
      "content": "Takahashi is playing sugoroku. The board has $N+1$ squares numbered $0$ to $N$. Takahashi starts at Square $0$ and head to Square $N$. In this sugoroku, we use a wheel showing numbers from $1$ through",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc189_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Takahashi is playing sugoroku.\nThe board has $N+1$ squares numbered $0$ to $N$. Takahashi starts at Square $0$ and head to Square $N$.\nIn this sugoroku, we use a wheel showing numbers from $1$ through...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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