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.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.util.StringTokenizer;

public class Main {
    int n;
    int[] ts, vs;
    double[] elapsedTs;

    public static void main(String args[]) {
        new Main().run();
    }

    void run() {
        FastReader sc = new FastReader();
        n = sc.nextInt();
        ts = new int[n];
        vs = new int[n];
        for (int i = 0; i < n; i++) {
            ts[i] = sc.nextInt();
        }
        for (int i = 0; i < n; i++) {
            vs[i] = sc.nextInt();
        }
        solve();
    }

    void solve() {
        elapsedTs = new double[n + 1];
        for (int i = 1; i <= n; i++) {
            elapsedTs[i] = elapsedTs[i - 1] + ts[i - 1];
        }
        double prevMaximumVelocity = 0.0;
        double area = 0.0;
        for (int i = 1; i <= 2* elapsedTs[n]; i++) {
            double maximumVelocity = maximumVelocity((double)i / 2, elapsedTs[n]);
            area += (maximumVelocity + prevMaximumVelocity) / 4;
            prevMaximumVelocity = maximumVelocity;
        }
        System.out.printf("%4f\n", area);
    }

    double maximumVelocity(double elapsedTime, double totalTime) {
        double[] constraints = new double[n + 2];
        constraints[n] = elapsedTime;
        constraints[n + 1] = totalTime - elapsedTime;
        for (int i = 0; i < n; i++) {
            if (elapsedTime < elapsedTs[i]) {
                constraints[i] = vs[i] + elapsedTs[i] - elapsedTime;
            } else if (elapsedTime > elapsedTs[i + 1]) {
                constraints[i] = vs[i] + elapsedTime - elapsedTs[i + 1];
            } else {
                constraints[i] = vs[i];
            }
        }
        double ret = Double.MAX_VALUE;
        for (int i = 0; i < constraints.length; i++) {
            ret = Math.min(ret, constraints[i]);
        }
        return ret;
    }

    static class FastReader {
        BufferedReader br;
        StringTokenizer st;

        public FastReader() {
            br = new BufferedReader(new
                    InputStreamReader(System.in));
        }

        String next() {
            while (st == null || !st.hasMoreElements())
            {
                try
                {
                    st = new StringTokenizer(br.readLine());
                }
                catch (IOException e)
                {
                    e.printStackTrace();
                }
            }
            return st.nextToken();
        }

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

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

        double nextDouble()
        {
            return Double.parseDouble(next());
        }

        String nextLine() {
            String str = "";
            try
            {
                str = br.readLine();
            }
            catch (IOException e)
            {
                e.printStackTrace();
            }
            return str;
        }
    }
}