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.StringTokenizer;

public class Main {
    public static void main(String[] args) throws IOException {
        FastScanner in = new FastScanner();
        int n = in.nextInt();
        int[] time = new int[n];
        for (int i = 0; i < n; i++) time[i] = in.nextInt();
        int[] maxV = new int[n + 1];
        for (int i = 0; i < n; i++) maxV[i] = in.nextInt();
        maxV[n] = 0;
        double distance = 0;
        double curV = 0;
        int tOld = 0;
        for (int i = 0; i < n; i++) {
            //Train will accelerate for some period of time, reach the cap for some period of time, and decelerate.
            int tNew = tOld; //tTerm = terminal velocity
            double tTerm = tOld + maxV[i] - curV;
            double accTime = tOld + time[i];
            for (int j = i + 1; j <= n; j++) { //When do we need to decelerate to reach this speed?
                tNew += time[j - 1];
                int newGoal = maxV[j]; //Need to be at newGoal by time tNew
                //curV + (tRes - tOld) - (tNew - tRes) = newGoal.  2 * tRes = newGoal + tOld + tNew - curV
                double tRes = -1 * curV;
                tRes += tOld + tNew + newGoal;
                tRes /= 2;
                if (tRes > tTerm) {
                    //curV + (tTerm - tOld) - (tNew - tRes) = newGoal.
                    //tRes = newGoal + tOld + tNew - curV - tTerm
                    tRes = -1 * curV + -1 *  tTerm;
                    tRes += newGoal + tOld + tNew;
                }
                
                if (tRes < accTime) accTime = tRes;
            }
            tNew = tOld + time[i];
            if (accTime > tTerm) {
                //Accelerating tOld through tTerm
                //No change tTerm through accTime
                //Decelerating accTime through tNew
                distance += curV * (tTerm - tOld) + (tTerm - tOld) * (tTerm - tOld) / 2;
                curV += tTerm - tOld;
                distance += curV * (accTime - tTerm);
                distance += curV * (tNew - accTime) - (tNew - accTime) * (tNew - accTime) / 2;
                curV -= tNew - accTime;
            } else {
                //Accelerating tOld through accTime
                //Decelerating accTime through tNew
                distance += curV * (accTime - tOld) + (accTime - tOld) * (accTime - tOld) / 2;
                curV += accTime - tOld;
                distance += curV * (tNew - accTime) - (tNew - accTime) * (tNew - accTime) / 2;
                curV -= tNew - accTime;
            }
            
        }
        System.out.println(distance);
    }

    public static class FastScanner {
        BufferedReader br;
        StringTokenizer st;

        public FastScanner(String s) {
            try {
                br = new BufferedReader(new FileReader(s));
            } catch (FileNotFoundException e) {
                // TODO Auto-generated catch block
                e.printStackTrace();
            }
        }

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

        String nextToken() {
            while (st == null || !st.hasMoreElements()) {
                try {
                    st = new StringTokenizer(br.readLine());
                } catch (IOException e) {
                    // TODO Auto-generated catch block
                    e.printStackTrace();
                }
            }
            return st.nextToken();
        }

        String nextLine() {
            st = null;
            try {
                return br.readLine();
            } catch (IOException e) {
                // TODO Auto-generated catch block
                e.printStackTrace();
            }
            return null;
        }

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

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

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