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;
using Int = long long;
signed main(){
  int n;
  cin>>n;
  vector<int> t(n),v(n);
  for(int i=0;i<n;i++) cin>>t[i],t[i]*=2;
  for(int i=0;i<n;i++) cin>>v[i],v[i]*=2;
  vector<vector<int> > dp(222,vector<int>(2,-1));
  dp[0][0]=0;
  int cur=0;
  for(int i=0;i<n;i++){
    for(int j=0;j<t[i];j++){
      bool f=cur%2;
      for(int k=0;k<222;k++) dp[k][!f]=-1;
      for(int k=0;k<=v[i];k++){
	if(dp[k][f]<0) continue;
	dp[k][!f]=max(dp[k][!f],dp[k][f]+k+k);
	if(k+1<=v[i])
	  dp[k+1][!f]=max(dp[k+1][!f],dp[k][f]+k+(k+1));
	if(k-1>=0)
	  dp[k-1][!f]=max(dp[k-1][!f],dp[k][f]+k+(k-1));
      }
      cur++;
    }
  }
  //assert(sum==cur);
  cout<<fixed<<setprecision(12)<<(double)dp[0][cur%2]/8.0<<endl;
  return 0;
}