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()))
v.append(0)
t.append(1)
sumoftimes=[0]
currentsum=0
for i in range(n+1):
  currentsum+=t[i]
  sumoftimes.append(currentsum)
ans=[]
for i in range(2*sumoftimes[n]+1):
  a=10**9
  for j in range(n+1):
    if 2*sumoftimes[j]<=i:
      if j>0:
        a=min(a,(i/2-sumoftimes[j]+v[j-1]))
    else:
      a=min(a,(sumoftimes[j]-i/2+v[j]))
  a=min(a,i/2)
  for j in range(n):
    if i<2*sumoftimes[j+1]:
      a=min(a,v[j])
      break
  if i==2*sumoftimes[n]:
    a=0
  ans.append(a)
finalans=0
for i in range(len(ans)-1):
  finalans+=(ans[i]+ans[i+1])/4
print(finalans)