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:
N=int(input())
t=list(map(int, input().split()))
v=list(map(int, input().split()))
T=sum(t)
V=[0]*(T+2)
from collections import deque
q=[deque() for i in range(101)]
s=0
for i in range(N):
    temp=min(V[s],v[i])
    q[temp].append(s)
    for j in range(t[i]):
        V[s+j+1]=v[i]
    s+=t[i]
q[0].append(T)
vans=[-1]*(T+1)
for i in range(101):
    while(len(q[i])>0):
        s=q[i].popleft()
        if vans[s]==-1:
            vans[s]=i
        if s>0 and vans[s-1]==-1:
            if V[s]>i:
                q[i+1].append(s-1)
            else:
                q[i].append(s-1)
        if s<T and vans[s+1]==-1:
            if V[s+1]>i:
                q[i+1].append(s+1)
            else:
                q[i].append(s+1)
ans=0
for i in range(T):
    if vans[i+1]>vans[i]:
        ans+=vans[i]+0.5
    elif vans[i+1]<vans[i]:
        ans+=vans[i+1]+0.5
    elif V[i+1]>vans[i]:
        ans+=vans[i]+0.25
    else:
        ans+=vans[i]
print(ans)