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Problem Statement

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 $

Sample Input 1

2 2 0

Sample 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 $1.5$.

Sample Input 2

2 2 1
1

Sample 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 $\sum_{i = 1}^{\infty} (i \times \frac{1}{2^i}) = 2$.

Sample Input 3

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

Sample Output 3

-1

Sample Input 4

100000 2 2
2997 92458

Sample Output 4

201932.2222