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=[int(i) for i in input().split()]
t_Table=[0]
for i in range(N):
  t_Table.append(t_Table[i]+t_List[i])
v_List=[int(i) for i in input().split()]
v_List.append(0)
Class=0
t=0
v=0
L=0
check=0
while t<=t_Table[-1]:
  for i in range(Class+1,N+1):
    if t_Table[i]!=t:
      if (v_List[i]-v)/(t_Table[i]-t)==-1:
        L+=v/2-1/8
        v+=-0.5
        check=-1
        break
  if check!=-1: 
    if v==v_List[Class]:
      L+=v/2
    else:
      L+=v/2+1/8
      v+=0.5
  t+=0.5
  if t_Table[Class+1]==t and Class<=N-2:
    Class+=1
    check=0
print(L)