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.*;

class Main {
	Scanner sc;
	int N, M;
	List<Integer>[] to;
	
	@SuppressWarnings("unchecked")
	private void calc() {
		sc = new Scanner(System.in);
		N = sc.nextInt();
		to = new ArrayList[N];
		for (int i = 0; i < N; i++)
			to[i] = new ArrayList<Integer>();
		
		M = sc.nextInt();
		for (int i = 0; i < M; i++) {
			int s = sc.nextInt()-1;
			int t = sc.nextInt()-1;
			to[s].add(t);
		}
		double min = Double.MAX_VALUE;
		// expected value
		double[] ev = new double[N];
		for (int i = 0; i < N-1; i++) {
			for (int j = N-2; j >= 0; j--) {
				double sum = 0d;
				double max = -Double.MAX_VALUE;
				for (int v : to[j]) {
					sum += ev[v]+1;
					max = Math.max(max, ev[v]);
				}
				int c = to[j].size();
				if (i == j && c > 1) ev[j] = (sum-max-1) / (c-1);
				else ev[j] = sum / c;
			}
			min = Math.min(min, ev[0]);
		}
		System.out.println(min);
	}
	
	public static void main(String[] args) {
		new Main().calc();
	}
}