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=[0]+[int(x) for x in str(input()).split()]+[0]
V=[0]+[int(x) for x in str(input()).split()]+[0]
 
time=0
t=[0]
for i in T:
  time+=i
  t.append(time)
ans=0
 
for j in range(0,2*time):
  m1=1000000000
  m2=1000000000
  t1=j/2
  t2=(j+1)/2
  for i in range(N+2):
    if t1<=t[i]:
      m1=min(m1,V[i]+(t[i]-t1))
    elif t1<=t[i+1]:
      m1=min(m1,V[i])
    else:
      m1=min(m1,V[i]+t1-t[i+1])
 
    if t2<=t[i]:
      m2=min(m2,V[i]+(t[i]-t2))
    elif t2<=t[i+1]:
      m2=min(m2,V[i])
    else:
      m2=min(m2,V[i]+t2-t[i+1])
    
  #print(1,m1,m2)
  ans+=0.25*(m1+m2)
print(ans)