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.util.*;

public class Main {
    private class Section {
        int l, r, v;

        public Section(int now, int t, int v) {
            this.l = now;
            this.r = now + t;
            this.v = v;
        }

        public double getMaxV(double t) {
            if (t < l) {
                return v + (l - t);
            } else if (r < t) {
                return v + (t - r);
            } else {
                return v;
            }
        }
    }

    public void main(Scanner sc) {
        int n = sc.nextInt();
        int t[] = new int[n];
        for (int i = 0; i < n; i++) {
            t[i] = sc.nextInt();
        }

        Section secs[] = new Section[n + 2];
        int sum = 0;
        for (int i = 1; i <= n; i++) {
            int v = sc.nextInt();
            secs[i] = new Section(sum, t[i - 1], v);
            sum += t[i - 1];
        }

        secs[0] = new Section(0, 0, 0);
        secs[n + 1] = new Section(sum, 0, 0);

        double ans = 0.0;
        for (double now = 0.0; now <= sum - 0.5; now += 0.5) {
            double v1 = secs[0].getMaxV(now);
            double v2 = secs[0].getMaxV(now + 0.5);
            for (int i = 1; i < n + 2; i++) {
                v1 = Math.min(v1, secs[i].getMaxV(now));
                v2 = Math.min(v2, secs[i].getMaxV(now + 0.5));
            }

            ans += (v1 + v2) * 0.25;
        }

        System.out.println(ans);
    }

    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        new Main().main(sc);
        sc.close();
    }
}