Cesar and Raul like betting and good food, in no particular order. They want to try out a new fancy restaurant and they decided to make a bet – they are going to play a game and the loser pays for dinner.

They have a box with $N$ balls. Each ball contains a distinct number between $1$ and $N$. Then, the game proceeds with these steps:
 - Initially, each person picks $C$ distinct numbers between $1$ and $N$ and writes them down on a separate card.
 - In each round, $D$ balls are drawn from the box uniformly at random. Cesar and Raul mark the ball numbers that appear on their respective card. The $D$ balls are then returned to the box.
 - The game stops when a player has marked all the numbers he chose. That player is the winner. If both players finish at the same time, it is a draw and they will split the dinner.

They are quite eager to try out this new restaurant and they’re now wondering: how many rounds will the game last until at least one of the players has marked all his numbers?

-----Task-----
Given the number $N$ of balls, the number $D$ of balls they draw from the box in each round, the amount $C$ of numbers in their cards and the numbers they wrote down, find the expected number of rounds the game will last.

-----Input-----
The first line of the input consists of three space separated integers: $N$, $D$, and $C$. $N$ is the number of balls, $D$ is the number of balls drawn in each round, and $C$ is the cards’ size. Each of the following two lines contains $C$ space separated integers: the numbers Cesar and Raul wrote down, respectively.

-----Constraints-----

$1$

$\leq $

$N$

$\leq $

$50$

Number of balls in the box

$1$

$\leq $

$D$

$\leq $

$\min (10, N)$

Number of balls drawn in each round

$1$

$\leq $

$C$

$\leq $

$\min (10, N)$

Cards’ sizeOutput

The output is the expected number of rounds of the game.

The result will be considered correct as long as the absolute error does not exceed $10^{-3}$.

-----Explanation for Sample Input 1-----
There are $2$ balls. Cesar picked number $1$ and Raul picked number $2$. In the first round, either number $1$ or $2$ will be drawn and so one of them wins right away.

-----Examples-----
Sample Input:
2 1 1
1
2
Sample Output:
1.00000000

Difficulty: competition

Starter code:

Solutions: