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.stream.*;
import java.util.*;

public class Main {
    static Scanner scanner = new Scanner(System.in);

    public static void main(String[]$) {
        int[] t = IntStream.range(0, scanner.nextInt()).map(i -> scanner.nextInt() * 2).toArray(), s = IntStream.concat(IntStream.of(0), IntStream.concat(Arrays.stream(t).flatMap(i -> Collections.nCopies(i, scanner.nextInt() * 2).stream().mapToInt(Integer::intValue)), IntStream.of(0))).toArray();
        IntStream.rangeClosed(0, s.length - 2).peek(i -> s[i + 1] = Math.min(s[i + 1], s[i] + 1)).map(i -> s.length - 2 - i).forEach(i -> s[i] = Math.min(s[i], s[i + 1] + 1));
        System.out.println(IntStream.rangeClosed(0, s.length - 2).mapToDouble(i -> Math.min(s[i], s[i + 1]) + Integer.compare(s[i + 1], s[i]) / 2d).sum() / 4);
    }
}