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()))
need=[(0,0)]*(N+1)
now=0
for i in range(N-1):
    now+=T[i]
    need[i+1]=(now,min(V[i],V[i+1]))
need[N]=(now+T[N-1],0)
TN=need[N][0]
res=[float('inf')]*(TN+1)
for i in range(TN+1):
    for j in range(N+1):
        if j>0:
            if need[j-1][0]<=i<=need[j][0]:
                res[i]=min(res[i],V[j-1])
        res[i]=min(res[i],need[j][1]+abs(need[j][0]-i))
ans=0
for i in range(TN):
    if abs(res[i]-res[i+1])==1:
        ans+=0.5*(res[i]+res[i+1])
    else:
        for j in range(N):
            if need[j][0]<=i and i+1<=need[j+1][0]:
                vv=V[j]
                break
        if vv>res[i]:
            ans+=res[i]+0.25
        else:
            ans+=res[i]
print(ans)