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 <cstdio>
using namespace std;
int n, s, mv, t[105], v[105], x[105], z[40005];
int main() {
	int i, j;
	cin>>n;
	for(i=1; i<=n; i++) cin>>t[i], t[i]*=2, x[i] = x[i-1] + t[i];;
	for(i=1; i<=n; i++) cin>>v[i], v[i]*=2;
	for(i=1, j=0; i<=n; i++) {
		z[j] = min(z[j], v[i]);
		for(j=j+1; j<=x[i]; j++) z[j] = min(z[j-1]+1, v[i]);
		j-=1;
	}
	for(i=x[n]-1; i; i--) {
		mv = min(z[i], mv+1);
		s += mv;
	}
	printf("%.9f", s/4.0);
	return 0;
}