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.List;
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();
		List<Integer>[] path = new ArrayList[n + 1];
		for(int i = 1; i < n + 1; i++) {
			path[i] = new ArrayList<>();
		}
		for(int i = 0; i < m; i++) {
			int s = sc.nextInt();
			int t = sc.nextInt();
			path[s].add(t);
		}
		double[] dp = new double[n + 1];
		double ans = Double.MAX_VALUE;
		for(int i = 1; i < n + 1; i++) {
			for(int j = n; j > 0; j--) {
				dp[j] = 0;
				double max = 0;
				if(j == n) {
					dp[j] = 0;
				} else {
					for(Integer to : path[j]) {
						dp[j] += dp[to] + 1;
						max = Math.max(max, dp[to] + 1);
					}
					if(i == j) {
						dp[j] -= max;
						if(path[j].size() - 1 != 0)
							dp[j] /= (path[j].size() - 1);
						else
							dp[j] = Double.MAX_VALUE;
					} else {
						dp[j] /= path[j].size();
					}
				}
			}

			ans = Math.min(ans, dp[1]);
		}
		System.out.println(ans);
	}

}