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 bisect import bisect_left, bisect_right
from itertools import accumulate

N = int(input())
ts = list(map(int, input().split()))
vs = list(map(int, input().split()))

Ts = list(accumulate([0] + ts))

# 横向きにカットするイメージ
def solve(begin, end, v):
    # print(' ' * v, begin, end, v, vs[bisect_right(Ts, begin) - 1:bisect_left(Ts, end)])
    if begin >= end:
        return 0
    if begin + 1 == end:
        if v < vs[bisect_right(Ts, begin) - 1]:
            return 0.25
        else:
            return 0
    min_v = min(vs[bisect_right(Ts, begin) - 1:bisect_left(Ts, end)])
    if min_v == v:
        ret = 0
        last = begin
        for i in range(bisect_right(Ts, begin) - 1, bisect_left(Ts, end)):
            if v == vs[i]:
                ret += solve(last, Ts[i], v)
                last = Ts[i + 1]
        ret += solve(last, end, v)
        return ret
    else:
        return end - begin - 1 + solve(begin + 1, end - 1, v + 1)

print(solve(0, Ts[-1], 0))