题目描述
In the year 2168, AtCoder Inc., which is much larger than now, is starting a limited express train service called AtCoder Express.
In the plan developed by the president Takahashi, the trains will run as follows:
- A train will run for (t_1 + t_2 + t_3 + ... + t_N) seconds.
- In the first t_1 seconds, a train must run at a speed of at most v_1 m/s (meters per second). Similarly, in the subsequent t_2 seconds, a train must run at a speed of at most v_2 m/s, and so on.
According to the specifications of the trains, the acceleration of a train must be always within ±1m/s^2. Additionally, a train must stop at the beginning and the end of the run.
Find the maximum possible distance that a train can cover in the run.
Constraints
- $1 \leq N \leq 100$
- $1 \leq t_i \leq 200$
- $1 \leq v_i \leq 100$
- All input values are integers.
Input
Input is given from Standard Input in the following format:
N
t_1 t_2 t_3 … t_N
v_1 v_2 v_3 … v_N
Output
Print the maximum possible that a train can cover in the run.
Output is considered correct if its absolute difference from the judge's output is at most 10^{-3}.
Examples
Input
1
100
30
Output
2100.000000000000000
Input
2
60 50
34 38
Output
2632.000000000000000
Input
3
12 14 2
6 2 7
Output
76.000000000000000
Input
1
9
10
Output
20.250000000000000000
Input
10
64 55 27 35 76 119 7 18 49 100
29 19 31 39 27 48 41 87 55 70
Output
20291.000000000000
输入格式
见题面
输出格式
见题面