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:
# AtCoder Beginner Contest 076
# D - AtCoder Express
# https://atcoder.jp/contests/abc076/tasks/abc076_d


def solve(T, V, velocity):
    accT = 1
    for t, max_v in zip(T, V):
        for i in range(accT, accT+2*t):
            velocity[i] = min(velocity[i], velocity[i-1]+0.5, max_v)
        accT += 2*t


def main():

    N, *A = map(int, open(0).read().split())
    T = A[:N]
    V = A[N:]

    INF = 10**10

    velocity = [INF]*(sum(T)*2+1)
    velocity[0] = 0
    velocity[-1] = 0
    solve(T, V, velocity)
    velocity = velocity[::-1]
    solve(T[::-1], V[::-1], velocity)

    ans = 0
    pre_v = velocity[0]
    for v in velocity[1:]:
        ans += (pre_v + v) * 0.25
        pre_v = v

    print(ans)


if __name__ == '__main__':
    main()