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:
import copy
N=int(input())
t=list(map(int,input().split()))
v=list(map(int,input().split()))

L=[0]
for i in range(N):
    a=[v[i] for j in range(t[i]*2)]
    #print(a)
    L+=a
X=[]
for i in range(1,len(L)):
    if L[i-1]>L[i]:
        X.append(i)
for i in X:
    L[i-1]=L[i]
L[-1]=0

S=copy.deepcopy(L)
#E=copy.deepcopy(L)
#print(L)
for i in range(1,len(L)):
    if L[i-1]<L[i]:
        for j in range(10**5):
            if S[i-1]+0.5*j>400 or (i-1+j)>(len(L)-1):
                break
            else:
                S[i-1+j]=min(S[i-1]+0.5*j,S[i-1+j])
    elif L[i-1]>L[i]:
        for j in range(10**5):
            if S[i]+0.5*j>400 or i<j:
                break
            else:
                S[i-j]=min(S[i]+0.5*j,S[i-j])
#print(S)
ans=0
for i in range(1,len(S)):
    ans+=(S[i-1]+S[i])/2
print(ans/2)