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:
import java.util.ArrayList;
import java.util.Scanner;

public class Main {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int N = sc.nextInt();
        int M = sc.nextInt();
        ArrayList<ArrayList<Integer>> passages = new ArrayList<ArrayList<Integer>>();
        for (int i = 0; i < N - 1; i++) {
            passages.add(new ArrayList<Integer>());
        }
        for (int i = 0; i < M; i++) {
            int s = sc.nextInt();
            int t = sc.nextInt();
            passages.get(s-1).add(t-1);
        }

        // no blocking
        double[] E = new double[N];
        for (int i = N - 2; i >= 0; i--) {
            double sum = 0;
            for (int dest: passages.get(i)) {
                sum += E[dest];
            }
            E[i] = sum / passages.get(i).size() + 1;
        }

        double minE = E[0];

        // with blocking
        for (int i = 0; i < N - 1; i++) {
            int len = passages.get(i).size();
            if (len == 1) {
                continue;
            }
            double maxE = 0;
            for (int dest: passages.get(i)) {
                maxE = Math.max(maxE, E[dest]);
            }
            E[i] = ((E[i] - 1) * len - maxE) / (len - 1) + 1;
            if (i > 0) {
                for (int j = i - 1; j >= 0; j--) {
                    double sum = 0;
                    for (int dest: passages.get(j)) {
                        sum += E[dest];
                    }
                    E[j] = sum / passages.get(j).size() + 1;
                }
            }
            minE = Math.min(minE, E[0]);
        }

        System.out.println(minE);
    }
}