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;
	vector<int> t(n), v(n);
	for (auto &i : t) {
		cin >> i;
	}
	for (auto &i : v) {
		cin >> i;
	}
	int m = 2 * accumulate(t.begin(), t.end(), 0) + 1;
	double a[m];
	for (int i = 0, s = 0; i < n; i++) {
		a[2 * s] = min(a[2 * s], double(v[i]));
		for (int j = 0; j < t[i]; j++) {
			a[2 * (s + j) + 1] = v[i];
			a[2 * (s + j) + 2] = v[i];
		}
		s += t[i];
	}
	a[0] = a[m - 1] = 0;
	for (int i = 1; i < m; i++) {
		a[i] = min(a[i], a[i - 1] + 0.5);
	}
	for (int i = m - 1; i; i--) {
		a[i - 1] = min(a[i - 1], a[i] + 0.5);
	}
	double ans = 0;
	for (int i = 0; i < m - 1; i++) {
		ans += (a[i] + a[i + 1]) / 4;
	}
	cout << fixed << setprecision(17) << ans << endl;
	return 0;
}