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())) + [0]

    CT = [0 for x in range(N+1)]  # cumulative time
    M = [float('inf') for x in range(len(T))]

    count = 0
    ct = 0
    for t, v in zip(T, V[:N]):
        CT[count] = ct
        ct += t
        count += 1
    CT[N] = ct

    for n in range(N):
        for i in range(n+1):
            M[i] = min(M[i], V[n+1] + (CT[n+1] - CT[i]))


    now = 0
    S = 0
    for q in range(len(T)):
        future = M[q]
        restriction = V[q]
        for s in range(T[q]*2):
            future -= 0.5
            pnow = now
            now = min(min(future, restriction), now + 0.5)
            S += (now + pnow)/4

    print(S)

if __name__ == "__main__":
    main()