Let N be a positive odd number.

There are $N coins, numbered 1, 2, \ldots, N$. For each $i (1 \leq i \leq N)$, when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

• N is an odd number.
• $1 \leq N \leq 2999$
• p_i is a real number and has two decimal places.
• 0 < p_i < 1

Input

Input is given from Standard Input in the following format:

N
$p_1 p_2 \ldots p_N$

Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80

Output

0.612

Input

1
0.50

Output

0.5

Input

5
0.42 0.01 0.42 0.99 0.42

Output

0.3821815872

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