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 sys
def input():
    return sys.stdin.readline()[:-1]
N=int(input())
t=list(map(int,input().split()))
v=list(map(int,input().split()))
l=[]
for i in range(N):
    l+=[v[i]]*t[i]
l.append(0)
T=sum(t)
for i in range(T-1,0,-1):
    if l[i]<l[i+1]:
        l[i+1]=l[i]+0.5
s=[100]*(T+1)
s[0],s[-1]=0,0
c=1
while c:
    c=0
    for i in range(1,T):
        if s[i]>s[i-1]+1:
            s[i]=s[i-1]+1
            c=1
        if s[i]>s[i+1]+1:
            s[i]=s[i+1]+1
            c=1
        if s[i]>l[i]:
            s[i]=int(l[i])
            c=1
a=0
for i in range(T):
    if s[i+1]!=s[i]:
        a+=(s[i+1]+s[i])/2
    else:
        if s[i]<l[i]:
            a+=s[i]+0.25
        else:
            a+=s[i]
print(a)