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

readline = sys.stdin.readline
MOD = 10 ** 9 + 7
INF = float('INF')
sys.setrecursionlimit(10 ** 5)


def main():
    from itertools import accumulate
    dh = 10

    n = int(readline())
    t_ = [0] + list(map(int, readline().split()))
    t_list = list(accumulate(t_))
    v_list = list(map(int, readline().split())) + [0]
    lim = [0] * (sum(t_list) * dh + 1)

    for i in range(n):
        st = dh * t_list[i]
        nt = dh * t_list[i + 1]
        cv = v_list[i]
        pv = lim[st]
        nv = v_list[i + 1]
        for ct in range(st, nt + 1):
            lim[ct] = min(cv, pv + (ct - st) / dh)

    for i in range(1, n + 1):
        st = dh * t_list[i]
        sv = v_list[i]
        for ct in range(st, -1, -1):
            lim[ct] = min(lim[ct], sv + (st - ct) / dh)

    ans = 0
    pv = 0
    for v in lim[1:]:
        ans += (pv + (v - pv) / 2) / dh
        pv = v

    print(ans)


if __name__ == '__main__':
    main()