Problem: 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

Solution (in PYTHON3):

Solution:
import sys
input = sys.stdin.readline
from collections import *

N = int(input())
t = list(map(int, input().split()))
v = list(map(int, input().split()))
end = [-1]*N
end[-1] = 0

for i in range(N-2, -1, -1):
    end[i] = min(end[i+1]+t[i+1], v[i], v[i+1])

ans = 0
s = 0

for i in range(N):
    e = end[i]

    if s+t[i]<=e:
        ans += (s+s+t[i])*t[i]/2
        s = s+t[i]
    else:
        up = min(v[i], (s+e+t[i])/2)
        ans += (s+up)*(up-s)/2
        ans += (e+up)*(up-e)/2
        ans += up*(t[i]-(up-s)-(up-e))
        s = e

print(ans)