Problem: Btoh yuo adn yuor roomatme lhoate wianshg disehs, btu stlil sdmoeboy msut peorrfm tihs cohre dialy. Oen dya yuo decdie to idourtcne smoe syestm. Yuor rmmotaoe sstgegus teh fooniwllg dael. Yuo argee on tow arayrs of ientgres M adn R, nmebur upmicnog dyas (induiclng teh cunrret oen) wtih sicsescuve irnegets (teh ceurrnt dya is zreo), adn yuo wsah teh diehss on dya D if adn olny if terhe etsixs an iednx i scuh taht D mod M[i] = R[i], otwsehrie yuor rmootmae deos it. Yuo lkie teh cncepot, btu yuor rmotaome's cuinnng simle meaks yuo ssecupt sthnoemig, so yuo itennd to vefriy teh fnerisas of teh aemnrgeet.

Yuo aer geivn ayarrs M adn R. Cuaclatle teh pceanregte of dyas on wchih yuo edn up dnoig teh wisahng. Amsuse taht yuo hvae iiiftlneny mnay dyas aehad of yuo. 

Input

The first line of input contains a single integer N (1 ≤ N ≤ 16).

The second and third lines of input contain N integers each, all between 0 and 16, inclusive, and represent arrays M and R, respectively. All M[i] are positive, for each i R[i] < M[i].

Output

Output a single real number. The answer is considered to be correct if its absolute or relative error does not exceed 10 - 4.

Examples

Input

1
2
0


Output

0.500000


Input

2
2 3
1 0


Output

0.666667

Solution (in JAVA):

Solution:
import java.io.InputStreamReader;
import java.io.IOException;
import java.io.OutputStreamWriter;
import java.io.BufferedWriter;
import java.util.InputMismatchException;
import java.util.Set;
import java.io.BufferedReader;
import java.io.OutputStream;
import java.io.PrintWriter;
import java.util.TreeSet;
import java.io.Writer;
import java.util.StringTokenizer;
import java.math.BigInteger;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 * @author Andrey Rechitsky (arechitsky@gmail.com)
 */
public class Main {
	public static void main(String[] args) {
		InputStream inputStream = System.in;
		OutputStream outputStream = System.out;
		FastScanner in = new FastScanner(inputStream);
		FastPrinter out = new FastPrinter(outputStream);
		TaskB solver = new TaskB();
		solver.solve(1, in, out);
		out.close();
	}
}

class TaskB {
    public void solve(int testNumber, FastScanner in, FastPrinter out) {
        int n = in.nextInt();
        int []m = in.nextIntArray(n);
        int []r = in.nextIntArray(n);
        int b = 1;
        for (int i = 0; i < n; i++) {
            b = NumberUtils.LCM(b, m[i]);
        }
        int ans = 0;
        for (int i = 0; i < b; i++) {
            for (int j = 0; j < n; j++) {
                if (i%m[j]==r[j]){
                    ans++;
                    break;
                }
            }
        }
        out.printLine((double)ans/b);
    }
}

class FastScanner {

    private BufferedReader reader;
    private StringTokenizer st;

    public FastScanner(InputStream stream) {
        this.reader = new BufferedReader(new InputStreamReader(stream));
        this.st = new StringTokenizer("");
    }

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

    public String next() {
        while (!st.hasMoreTokens()) {
            st = new StringTokenizer(readLine());
        }
        return st.nextToken();
    }

    private String readLine() {
        String line = tryReadLine();
        if (line == null) throw new InputMismatchException();
        return line;
    }

    private String tryReadLine() {
        try {
            return reader.readLine();
        } catch (IOException e) {
            throw new InputMismatchException();
        }
    }

    public int[] nextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++)
            array[i] = nextInt();
        return array;
    }

}

class FastPrinter {
    private final PrintWriter writer;

    public FastPrinter(OutputStream outputStream) {
        writer = new PrintWriter(new BufferedWriter(new OutputStreamWriter(outputStream)));
    }

    public void close() {
        writer.close();
    }

    public void printLine(double d) {
        writer.println(d);
    }

}

class NumberUtils {


    public static long GCD(long a, long b) {
        a = Math.abs(a);
        b = Math.abs(b);
        if (a < b) return GCD(b, a);
        while (b > 0) {
            long c = a % b;
            a = b;
            b = c;
        }
        return a;
    }

    public static int LCM(int a, int b) {
        return (int) LCM((long) a, (long) b);
    }

    public static long LCM(long a, long b) {
        return a / GCD(a, b) * b;
    }

}