Coins

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

inputFormat

Input

Input is given from Standard Input in the following format:

N p_1 p_2 \ldots p_N

outputFormat

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

样例

5
0.42 0.01 0.42 0.99 0.42

0.3821815872