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(){
 long n;
   cin >> n;
   
  long t[n+1],v[n+1];
  t[0]=0;
  v[0]=0;
  for(long i=1;i<=n;i++) {
    cin >> t[i];
    t[i]+=t[i-1];
  }
  for(long i=1;i<=n;i++) cin >> v[i];

  long res=0;
  long right=0;
  for(long i=1;i<2*t[n]+1;i++){
   long left=right;
   right=min(i,2*t[n]-i);
    for(long j=1;j<=n;j++){
     if(i<=2*t[j-1]) right=min(right,2*v[j]+2*t[j-1]-i);
     else if(2*t[j]<=i) right=min(right,2*v[j]+i-2*t[j]);
     else right=min(right,2*v[j]);
    }
    res+=(left+right);
  }
  
   cout << setprecision(12) << double(res)/8.0 << endl;
}