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:

def spaceinput():
    return list(map(int,input().split(" ")))

def interval(t):
    pass

N=int(input())
T=spaceinput()
v=spaceinput()

Tmax=0

for i in range(N):
    Tmax+=(T[i])
#mv=[0]*(Tmax+1)
mv={}

frange=[x for x in range(2*Tmax+1)]

k=0
sT=T[0]
d=0.5

mv[0]=0
for t in [x/2 for x in range(1,2*Tmax)]:

    if sT<t:
        k+=1
        if k<len(T):
            sT+=T[k]
        else:
            break

    mv[t]=min(v[k],mv[t-0.5]+d)

mv[Tmax]=0

k=N-1
sT=Tmax-T[-1]
for t in reversed([x/2 for x in range(1,2*Tmax)]):
    if sT>t:
        k-=1
        sT-=T[k]

    mv[t]=min(v[k],mv[t],mv[t+d]+d)


res=0
k=0
sT=T[0]
for t in [x/2 for x in range(1,2*Tmax)]:
    if sT<t:
        if t==Tmax-1:
            break
        k+=1
        if k<len(T):
            sT+=T[k]
    res+=mv[t]*0.5

print(res)