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.*;
import java.util.*;

public class Main {

    private static class InputReader {
        private BufferedReader reader;
        private StringTokenizer tokenizer;

        public InputReader(InputStream stream) {
            reader = new BufferedReader(new InputStreamReader(stream), 32768);
            tokenizer = null;
        }

        public String next() {
            while (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (IOException exp) {
                    throw new RuntimeException(exp);
                }
            }
            return tokenizer.nextToken();
        }

        public int nextInt() {
            return Integer.parseInt(next());
        }

        public long nextLong() {
            return Long.parseLong(next());
        }
    }

    public static void main(String[] args) {    //throws FileNotFoundException {
        InputStream inputStream = System.in;    //new FileInputStream("input.txt");
        OutputStream outputStream = System.out; //new FileOutputStream("output.txt");
        InputReader in = new InputReader(inputStream);
        PrintWriter out = new PrintWriter(outputStream);

        Task newTask = new Task();
        newTask.solve(in, out);

        out.close();
    }

    private static class Task {
        //.constant
        private final int INF = Integer.MAX_VALUE;
        private final int MOD = (int)1e9 + 7;

        //.data
        private int[] t = new int[101];
        private int[] v = new int[101];
        private int[] maxv = new int[101];

        //.code
        public void solve(InputReader in, PrintWriter out) {
            int n = in.nextInt();
            for (int i = 0; i < n; ++i) {
                t[i] = in.nextInt();
            }
            for (int i = 0; i < n; ++i) {
                v[i] = in.nextInt();
            }

            for (int i = n - 1; i > 0; --i) {
                int minV = Math.min(v[i - 1], v[i]);
                maxv[i] = minV - maxv[i + 1] <= t[i] ? minV : maxv[i + 1] + t[i];
            }

            double s = 0;
            double curV = 0;
            for (int i = 0; i < n; ++i) {
                double nextV = Math.min(curV + t[i], maxv[i + 1]);
                double diffV = Math.abs(nextV - curV);
                double c = Math.abs(t[i] - diffV);
                s += c * Math.max(curV, nextV) + diffV * Math.min(curV, nextV) + diffV * diffV / 2.0;
                if (c / 2.0 + Math.max(curV, nextV) > v[i]) {
                    double h = v[i] - Math.max(curV, nextV);
                    s += h * (c - h);
                } else {
                    s += c * c / 4.0;
                }
                curV = nextV;
            }

            out.println(s);

        }//end

    }

}