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.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.InputMismatchException;
import java.io.IOException;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 *
 * @author Pradyumn
 */
public class Main {
    public static void main(String[] args) {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        FastReader in = new FastReader(inputStream);
        PrintWriter out = new PrintWriter(outputStream);
        TaskD solver = new TaskD();
        solver.solve(1, in, out);
        out.close();
    }

    static class TaskD {
        public void solve(int testNumber, FastReader in, PrintWriter out) {
            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;
                }

            }
            out.printf("%.15f\n", distance);
        }

    }

    static class FastReader {
        private InputStream stream;
        private byte[] buf = new byte[8192];
        private int curChar;
        private int pnumChars;
        private FastReader.SpaceCharFilter filter;

        public FastReader(InputStream stream) {
            this.stream = stream;
        }

        private int pread() {
            if (pnumChars == -1) {
                throw new InputMismatchException();
            }
            if (curChar >= pnumChars) {
                curChar = 0;
                try {
                    pnumChars = stream.read(buf);
                } catch (IOException e) {
                    throw new InputMismatchException();
                }
                if (pnumChars <= 0) {
                    return -1;
                }
            }
            return buf[curChar++];
        }

        public int nextInt() {
            int c = pread();
            while (isSpaceChar(c))
                c = pread();
            int sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = pread();
            }
            int res = 0;
            do {
                if (c == ',') {
                    c = pread();
                }
                if (c < '0' || c > '9') {
                    throw new InputMismatchException();
                }
                res *= 10;
                res += c - '0';
                c = pread();
            } while (!isSpaceChar(c));
            return res * sgn;
        }

        private boolean isSpaceChar(int c) {
            if (filter != null) {
                return filter.isSpaceChar(c);
            }
            return isWhitespace(c);
        }

        private static boolean isWhitespace(int c) {
            return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
        }

        private interface SpaceCharFilter {
            public boolean isSpaceChar(int ch);

        }

    }
}