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())
tlist=list(map(int,input().split()))
vlist=list(map(int,input().split()))

def const(l,r,v,t):
  if t<l:
    return v+(l-t)
  elif r<t:
    return v+(t-r)
  else:
    return v
  
const_list=[(0,0,0)]
tsum=0
for i in range(N):
  l=tsum
  r=tsum+tlist[i]
  v=vlist[i]
  const_list.append((l,r,v))
  tsum+=tlist[i]
const_list.append((tsum,tsum,0))
#print(const_list)

answer=0
t=0
while(t<tsum):
  s1=s2=10**9
  for con in const_list:
    l,r,v=con
    s1=min(s1,const(l,r,v,t))
    s2=min(s2,const(l,r,v,t+0.5))
  #print(s1,s2)
    
  answer+=(s1+s2)*0.5/2
  t+=0.5
  
print(answer)