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.PrintWriter;
import java.util.Scanner;

public class Main {
    static Scanner sc = new Scanner(System.in);
    static PrintWriter out = new PrintWriter(System.out);

    public static final void main(String[] args) {
        new Main().solve();
        out.flush();
    }

    // ========================================================================= //

    public void solve() {
        int n = sc.nextInt();
        double[] t = new double[n];
        double[] v = new double[n + 1];
        for (int i = 0; i < n; i++) t[i] = sc.nextDouble();
        for (int i = 0; i < n; i++) v[i] = sc.nextDouble();
        v[n] = 0.0;

        double ans = 0.0, curV = 0.0;
        for (int i = 0; i < n; i++) {
            for (double j = 0; j < t[i]; j+=0.5) {
                boolean up = true, eq = true;
                double  tc = t[i] - j;
                for (int k = i + 1; k < n + 1; k++) {
                    if (curV + 0.5 - tc >= v[k]) up = false;
                    if (curV - tc >= v[k]) eq = false;
                    if (k < n) tc += t[k];
                }

                if (up && curV + 0.5 <= v[i]) {
                    ans += curV * 0.5 + 0.125;
                    curV += 0.5;
                }
                else if (eq) {
                    ans += curV * 0.5;
                }
                else {
                    ans += curV * 0.5 - 0.125;
                    curV -= 0.5;
                }
            }
        }
        out.println(ans);
    }
}