In a competition for students, students are asked to collect some paint and fill them into buckets. Buckets are on a rail system and they can take different amount of paints. On the rail system they wait certain time and then, they leave. The gathered paint at that time can be poured into the active bucket. If it's full, the half of the extra paint can be poured to the next and the other half of the extra paint can be poured to the previous buckets. If they are also full (or they overflood), the gathered extra paint is wasted out.

Question:
Write a program that is going to first read bucket sizes and then collected paint amounts. The program is going to find out the amount of paint collected by this team.
Note: The first bucket doesn't have the previous and the last bucket doen't have the next. Thus, if the first one overfloods to the previous, the extra paint is wasted.

Input specification
You will be first given an integer number (n) which is number of buckets where 0 ≤ n ≤ 15000. Then, you will be given n bucket size in the following line. The buckets can have sizes between 1 and 20litters. The following line will have n floating point numbers, the amount of paint gathered by students which may vary between 0 and 20.

Output specification
Show a floating point number (with 3 decimal place precision), the amount of paint gathered bu students.

Explanation:
They always pour the collected paint firstly to the active bucket. If the bucket is full and there is extra paint, half of the remaining paint is poured to the bucket on the right and the other half is poured to the bucket on the left.

The size of first bucket is 5 liters. And the students have collected 7 liters. So, they pour 5 liters to the first bucket. Half of remaining amount (1 liter) is poured to the second bucket. The capacity of the second bucket is 14 liters. And students have collected 7.5 liters. So, they can pour all of it to the second bucket. The capacity of the third bucket is also 14 liters. And students have collected 20 liters. They can firstly fill the third bucket (14 liters), then half of the remaining amount can be poured to the second bucket and the other half can be poured to the fourth bucket. Thus, the second bucket has 11.5 liters of paint. (1 liter from the first bucket, 7.5 liters from the second bucket and 3 liters from the third bucket.)