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 JAVA):

Solution:
import java.io.BufferedReader;
import java.io.InputStreamReader;

public class Main {
	public static void main(String[] args) throws Exception {
		BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
		int n = Integer.parseInt(br.readLine());

		int[] t = new int[n + 2];
		String[] sa = br.readLine().split(" ");
		for (int i = 0; i < n; i++) {
			t[i + 1] = t[i] + Integer.parseInt(sa[i]);
		}
		t[n + 1] = t[n];

		int[] v = new int[n + 2];
		sa = br.readLine().split(" ");
		for (int i = 0; i < n; i++) {
			v[i + 1] = Integer.parseInt(sa[i]);
		}
		br.close();

		double ans = 0;
		double pre = 0;
		for (double x = 0.5; x <= t[n]; x+=0.5) {
			double min = Double.MAX_VALUE;
			for (int i = 0; i < v.length; i++) {
				min = Math.min(min, calc(t, v, i, x));
			}
			ans += (min + pre) / 4;
			pre = min;
		}
		System.out.println(ans);
	}

	static double calc(int[] t, int[] v, int i, double x) {
		if (i == 0) {
			return x;
		}
		if (x < t[i - 1]) {
			return v[i] + t[i - 1] - x;
		} else if (t[i - 1] <= x && x <= t[i]) {
			return v[i];
		} else {
			return v[i] + x - t[i];
		}
	}
}