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:
from itertools import chain
 
double = lambda c: int(c)*2
 
N = int(input())
T = tuple(map(double,input().split()))
V = tuple(map(double,input().split()))
 
dp = [0]
 
for t,v_max,vf_max in zip(T,V,chain(V[1:],(0,))):

  ndp = [None]*(min(vf_max,v_max)+1)
  
  for vf in range(len(ndp)):
    vi = min(len(dp)-1,vf+t)
    while vi >= 0 and dp[vi] is None:
      vi -= 1

    if vi < max(0,vf-t):
      ndp[vf] = None
    else:
      vp = (vf+t+vi)/2
      ndp[vf] = dp[vi] + 2*vp*vp-vf*vf-vi*vi - 2*max(0,vp-v_max)**2
  dp = ndp
 
print(dp[0]/8)