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:
"""
ABC076_D AtCoderExpress
まず区間に注目してEditorialみたいなグラフを思い描くことがきつかった。
そのうえで、区間Limitをバグなく実装するのが結構ポイント。
あとはVVあたりがバグってないかをPrintしながら見ていく形。
"""
N = int(input())
T = list(map(int,input().split()))
V = list(map(int,input().split()))
TMAX = sum(T)

TS = [0] * (N+1)
for i in range(N):
  TS[i+1] = TS[i] + T[i]
T_Left = TS[:N]
T_Right = TS[1:]
#print(T_Left,T_Right,V)

## これで下記がちゃんとつくれた。
# Tleft[i] <= t <= Tright[i] の間、 V[i]以下でなければいけない

# VV[tt] tt/2秒時の速度。両端がゼロ、という条件から初期値を生成する。
VV = [min(i*0.5, (TMAX*2-i) * 0.5) for i in range(TMAX*2+1)]
#print(VV)

# 区間iについて
for i in range(N):
  for tt in range(TMAX*2+1):
    VV[tt] = min(VV[tt], V[i] + max(0, (T_Left[i]*2 - tt) * 0.5, (tt-T_Right[i]*2) * 0.5))
#print(VV)
# 面積は両端がゼロであることから、速度合計を２で割る（0.5秒ごとだから）で出る。
# v-tグラフの面積を台形の面積で考えればよい。
print(sum(VV)/2)