Problem: There is a cave consisting of N rooms and M one-directional passages. The rooms are numbered 1 through N.

Takahashi is now in Room 1, and Room N has the exit. The i-th passage connects Room s_i and Room t_i (s_i < t_i) and can only be traversed in the direction from Room s_i to Room t_i. It is known that, for each room except Room N, there is at least one passage going from that room.

Takahashi will escape from the cave. Each time he reaches a room (assume that he has reached Room 1 at the beginning), he will choose a passage uniformly at random from the ones going from that room and take that passage.

Aoki, a friend of Takahashi's, can block one of the passages (or do nothing) before Takahashi leaves Room 1. However, it is not allowed to block a passage so that Takahashi is potentially unable to reach Room N.

Let E be the expected number of passages Takahashi takes before he reaches Room N. Find the value of E when Aoki makes a choice that minimizes E.

Constraints

* 2 \leq N \leq 600
* N-1 \leq M \leq \frac{N(N-1)}{2}
* s_i < t_i
* If i != j, (s_i, t_i) \neq (s_j, t_j). (Added 21:23 JST)
* For every v = 1, 2, ..., N-1, there exists i such that v = s_i.

Input

Input is given from Standard Input in the following format:


N M
s_1 t_1
:
s_M t_M


Output

Print the value of E when Aoki makes a choice that minimizes E. Your output will be judged as correct when the absolute or relative error from the judge's output is at most 10^{-6}.

Examples

Input

4 6
1 4
2 3
1 3
1 2
3 4
2 4


Output

1.5000000000


Input

3 2
1 2
2 3


Output

2.0000000000


Input

10 33
3 7
5 10
8 9
1 10
4 6
2 5
1 7
6 10
1 4
1 3
8 10
1 5
2 6
6 9
5 6
5 8
3 6
4 8
2 7
2 9
6 7
1 2
5 9
6 8
9 10
3 9
7 8
4 5
2 10
5 7
3 5
4 7
4 9


Output

3.0133333333

Solution (in JAVA):

Solution:
//package atcoder.beginner144;

import java.io.*;
import java.util.*;

public class Main {
    private static List<Integer>[] paths;
    private static double[] dp;
    private static int N, M;
    private static double ans;

    public static void main(String[] args) {
//        try {
//            FastScanner in = new FastScanner(new FileInputStream("src/input.in"));
//            PrintWriter out = new PrintWriter(new FileOutputStream("src/output.out"));
        FastScanner in = new FastScanner(System.in);
        PrintWriter out = new PrintWriter(System.out);

            solve(in, out);

//        }
//        catch (IOException e) {
//            e.printStackTrace();
//        }
    }

    private static void input(FastScanner in, PrintWriter out) {
        N = in.nextInt();
        M = in.nextInt();
        paths = new List[N + 1];
        dp = new double[N + 1];

        for(int i = 1; i <= N; i++) {
            paths[i] = new ArrayList<>();
        }
        for(int i = 0; i < M; i++) {
            int s = in.nextInt(), t = in.nextInt();
            paths[s].add(t);
        }
    }

    private static void f(int block) {
        dp[N] = 0;
        for(int i = N - 1; i >= 1; i--) {
            double sum = 0.0;
            double maxAdj = 0.0;
            int num = paths[i].size();
            for(int j : paths[i]) {
                sum += dp[j];
                maxAdj = Math.max(maxAdj, dp[j]);
            }
            if(i == block && num > 1) {
                num--;
                sum -= maxAdj;
            }
            dp[i] = sum / num + 1;
        }
        ans = Math.min(ans, dp[1]);
    }

    private static void solve(FastScanner in, PrintWriter out) {
        input(in, out);
        ans = Double.MAX_VALUE;
        for(int i = 1; i < N; i++) {
            f(i);
        }
        out.println(ans);
        out.close();
    }

    static class FastScanner {
        BufferedReader br;
        StringTokenizer st;

        FastScanner(InputStream stream) {
            try {
                br = new BufferedReader(new InputStreamReader(stream));
            } catch (Exception e) {
                e.printStackTrace();
            }
        }

        String next() {
            while (st == null || !st.hasMoreTokens()) {
                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;
        }
    }
}