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.Scanner;

public class Main {
  public static void main(String[] args) {
    Scanner scanner = new Scanner(System.in);

    int N = scanner.nextInt();
    int[] t = new int[N + 1];
    int[] v = new int[N + 2];

    for (int i = 0; i < N; i++) t[i + 1] = scanner.nextInt() * 2 + t[i];
    for (int i = 0; i < N; i++) v[i + 1] = scanner.nextInt() * 2;

    long dist = 0;
    long cv = 0;
    for (int i = 1; i <= N; i++) {
      for (int j = t[i - 1] + 1; j <= t[i]; j++) {
        long f = v[i];
        for (int k = 0; k < i; k++) f = Math.min(f, v[k] + j - t[k]);
        for (int k = i + 1; k <= N + 1; k++) f = Math.min(f, v[k] + t[k - 1] - j);
        dist += cv + f;
        cv = f;
      }
    }

    System.out.println(dist / 8.d);
  }
}