Anthony and Cora play a very exciting game. Initially,
Anthony has $N$ points and
Cora has $M$ points. The
game then goes on for several rounds, during each round, either
Anthony wins or Cora wins, the loser of the round loses
$1$ point while nothing
happens to the winner of the round. The game ends when either
Anthony or Cora has no points left, and the one still left with
points is the winner of the game.
Cora promises Anthony a sweet prize if he wins the game, but
will cruelly humiliate Anthony if he loses. Anthony can very
accurately evaluate himself and perfectly predict that his
probability of winning the round $i$ is exactly $p_ i$. Now, in order to decide
whether to play this game with Cora or not, he needs to know
the probability of him winning the game.
Help Anthony find his probability of winning!
Input
The first line contain integers $1\leq N,M\leq 1\, 000$. $N+M1$ lines follow, with the
$i$th line containing
$0\leq p_ i\leq 1$,
$p_ i$ has at most
$6$ decimal digits.
Output
Output a single line containing the probability of Anthony
winning this game. Your answer will be considered correct if
its absolute or relative error doesnâ€™t exceed $10^{6}$.
Sample Input 1 
Sample Output 1 
1 1
0.500000

0.500000

Sample Input 2 
Sample Output 2 
3 2
1.000000
0.000000
1.000000
0.000000

1.000000
