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.InputStream;
import java.io.InputStreamReader;
import java.util.*;

public class Main {

    static int N;
    static int[] T;
    static int[] V;

    public static void main(String[] args) {
        FastScanner sc = new FastScanner(System.in);
        N = sc.nextInt();
        T = sc.nextIntArray(N);
        V = sc.nextIntArray(N);
        System.out.println( solve() );
    }

    private static double solve() {
        List<Restriction> rs = mkRestriction();
        int totalTime = 0;
        for (int i = 0; i < N; i++) {
            totalTime += T[i] * 2;
        }

        // speed, time は二倍に拡大している
        int currentSpeed = 0;
        double run = 0;
        for (int t = 0; t < totalTime; t++) {
            // t = 0 -> 0 to 1の間の出来事
            int nextLimit = calcMinLimit(t+1, rs);

            if( nextLimit == currentSpeed ) {
                run += currentSpeed;

            } else if( nextLimit > currentSpeed ) {
                run += currentSpeed + 0.5;
                currentSpeed++;

            } else {
                run += currentSpeed - 0.5;
                currentSpeed--;
            }
        }

        return run * 0.25;
    }

    private static int calcMinLimit(int t, List<Restriction> rs) {
        int min = Integer.MAX_VALUE;
        for (Restriction r : rs) {
            min = Math.min(r.limitAt(t), min);
        }
        return min;
    }

    private static List<Restriction> mkRestriction() {
        List<Restriction> rs = new ArrayList<>();
        rs.add(new Restriction(0, 0, 0));
        int start = 0;
        for (int i = 0; i < N; i++) {
            int end = start + T[i] * 2;
            int speed = V[i] * 2;
            rs.add( new Restriction(start, end, speed) );

            start = end;
        }
        rs.add(new Restriction(start, start, 0));
        return rs;
    }

    private static class Restriction {
        private int start;
        private int end;
        private int speed;

        public Restriction(int start, int end, int speed) {
            this.start = start;
            this.end = end;
            this.speed = speed;
        }

        public int limitAt(int t) {
            if( t < start ) {
                return speed + start - t;
            } else if( t > end ){
                return speed + t - end;
            } else {
                return speed;
            }
        }
    }

    static class FastScanner {

        private BufferedReader reader;
        private StringTokenizer tokenizer;

        FastScanner(InputStream in) {
            reader = new BufferedReader(new InputStreamReader(in));
            tokenizer = null;
        }

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

        String nextLine() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    return reader.readLine();
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }

            return tokenizer.nextToken("\n");
        }

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

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

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

        int[] nextIntArray(int n) {
            int[] a = new int[n];
            for (int i = 0; i < n; i++)
                a[i] = nextInt();
            return a;
        }

        long[] nextLongArray(int n) {
            long[] a = new long[n];
            for (int i = 0; i < n; i++)
                a[i] = nextLong();
            return a;
        }
    }
}