Predicting a Probability

The late-delivery model is fitted, and now the dispatch screen has to run it. A job is described by several numbers — distance, parcels, how far behind the driver already is — and the model holds one weight per number plus a starting amount that applies before any of them are read. Those combine into the unbounded score, and the screen shows the probability that score corresponds to, using the same standard sigmoid curve on the natural exponential as before.

Task: Print the probability the model gives this job.

Input

The first line holds one integer k, how many numbers describe a job, where 1 <= k <= 50. The second line holds k + 1 numbers separated by single spaces: the starting amount first, then the k weights. The third line holds the job's k numbers separated by single spaces, in the same order as the weights.

Output

One line holding the probability rounded to 4 decimal places, with no label.

Example:

Input:
2
0 1 1
1 1

Output:
0.8808