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 accumulate

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

# 各区間の制限速度条件のみを満たす最大速度のグラフを求める
accTs = list(accumulate(ts))
sumT = accTs[-1]
vMaxs = [[] for i in range(N + 2)]
for i, (v, l, r) in enumerate(zip(vs + [0, 0], [0] + accTs[:-1] + [0, sumT], accTs + [0, sumT])):

    vms = [v + l - t / 2 for t in range(0, l * 2)] + \
          [v] * ((r - l) * 2 + 1) + \
          [v + t / 2 - r for t in range(r * 2 + 1, sumT * 2 + 1)]

    vMaxs[i] = vms

# 最小値がその時刻における列車の速度になる
vMaxs = list(map(min, *vMaxs))

print(sum(vMaxs) / 2)