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 CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;

int main(){
  int N;
  cin>>N;
  std::vector<double> t(N),v(N);
  for(int i=0;i<N;i++)cin>>t[i];
  for(int i=0;i<N;i++)cin>>v[i];
  double ts=0;
  for(int i=0;i<N;i++)ts+=t[i];
  std::vector<double> lo(ts*2+1);
  for(double i=0;i<2*ts+1;i++)lo[i]=i/2;
  for(double i=0;i<2*ts+1;i++)lo[2*ts-i]=min(lo[i],i/2);
  double tnow=0;
  for(double i=0;i<N;i++){
    for(double j=0;j<2*ts+1;j++){
      if(j<tnow)lo[j]=min(lo[j],v[i]+(tnow-j)/2);
      if(j>=tnow&&j<=tnow+t[i]*2)lo[j]=min(lo[j],v[i]);
      if(j>tnow+t[i]*2)lo[j]=min(lo[j],v[i]+(j-tnow-t[i]*2)/2);
    }
    tnow+=t[i]*2;
  }
  double ans=0;
  for(double i=1;i<ts*2+1;i++)ans+=(lo[i]+lo[i-1])/4;
  //for(double i=0;i<ts*2+1;i++)cout<<lo[i]<<endl;
  cout<<fixed<<setprecision(12)<<ans<<endl;
}