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:
INF=1<<30
N = int(input())
t = [int(_) for _ in input().split()]
st=[0]*(N+1)
for i in range(N):
    st[i+1]=t[i]+st[i] # st[i] i番目の区間の終了時刻、ただし0番目の区間は時刻0に終了したものとする
T=st[N]
st.append(T)
v = [int(_) for _ in input().split()] # i番目の区間の制限速度
v.insert(0,0)
v.append(0)
l =[0]

now=0.5
ind = 1 # 現在いる区間の番号

while now < T:
    v1=INF
    v2=INF
    for j in range(ind):
        v1=min(v1,v[j]+now-st[j])
    for j in range(ind,N+1):
        v2=min(v2,v[j+1]-now+st[j])
    l.append(min(v[ind],v1,v2))
    now+=0.5
    if now>=st[ind]:
        ind+=1

l.append(0)
res=0

for i in range(len(l)):
    res+=l[i]/2

print(res)