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:
def main():
    N = int(input())
    T = list(map(int, input().split()))
    V = list(map(int, input().split()))
    C = [0] * (N+1)
    for i in range(N-1):
        C[i+1] = min(V[i], V[i+1])
    for _ in range(N):
        for i in range(N):
            left = C[i]
            right = C[i+1]
            C[i] = min(C[i], right + T[i])
            C[i+1] = min(C[i+1], left + T[i])
    ans = 0
    for i in range(N):
        if (V[i] - C[i]) + (V[i] - C[i+1]) < T[i]:
            lx = V[i] - C[i]
            rx = V[i] - C[i+1]
            ans += (C[i] + V[i]) * lx / 2 + \
                (C[i+1] + V[i]) * rx / 2 + \
                V[i] * (T[i]-lx-rx)
        else:
            M = max(C[i], C[i+1])
            m = min(C[i], C[i+1])
            x = (T[i] - (M-m)) / 2
            ans += (M + M+x) * x / 2 + (m + M+x) * (x+(M-m)) / 2
    print(ans)


if __name__ == "__main__":
    main()