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<iostream>
#include<algorithm>
#include<iomanip>
using namespace std;
int n;
double t[110],v[110],sum[110];
int main()
{
	cin>>n;
	for(int i=0;i<n;i++)
	{
		cin>>t[i+1];
		sum[i+1]=sum[i]+t[i+1];
	}
	for(int i=0;i<n;i++)cin>>v[i+1];
	double ans=0;
	double p=0;
	int f=0;
	for(double i=0.5;i<=sum[n];i+=0.5)
	{
		if(i>sum[f])f++;
		double now=v[f];
		for(int j=0;j<=n+1;j++)
		{
			double c=now;
			if(j<f)
			{
				c=v[j]+i-sum[j];
			}
			else if(j>f)
			{
				c=v[j]+sum[j-1]-i;
			}
			if(now>c)now=c;
		}
		ans+=(p+now)/4.0;
		p=now;
	}
	cout<<fixed<<setprecision(9)<<ans<<endl;
}