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:
from itertools import chain

def main():
    N = int(input())
    t = list(map(int, input().split()))
    v = list(map(int, input().split()))

    vmax = list(chain.from_iterable([i] * (2 * j) for i, j in zip(v, t)))
    vmax = [min(v1,v2) for v1, v2 in zip([0] + vmax, vmax + [0])]

    now = 0
    for i in range(1,len(vmax)):
        vmax[i] = min(vmax[i], now + 0.5)
        now = vmax[i]
    
    now = 0
    for i in range(len(vmax)-1,0,-1):
        vmax[i] = min(vmax[i], now + 0.5)
        now = vmax[i]

    ans = 0
    for i in range(len(vmax)-1):
        if vmax[i] == vmax[i+1]:
            ans += 0.5 * vmax[i]
        else:
            ans += (vmax[i] + vmax[i+1]) * 0.5 / 2

    print(ans)


if __name__ == "__main__":
    main()