Problem:
Score : 600 points

Problem Statement

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}.

Sample Input 1

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

Sample Output 1

1.5000000000

If Aoki blocks the passage from Room 1 to Room 2, Takahashi will go along the path 1 → 3 → 4 with probability \frac{1}{2} and 1 → 4 with probability \frac{1}{2}. E = 1.5 here, and this is the minimum possible value of E.

Sample Input 2

3 2
1 2
2 3

Sample Output 2

2.0000000000

Blocking any one passage makes Takahashi unable to reach Room N, so Aoki cannot block a passage.

Sample Input 3

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

Sample Output 3

3.0133333333

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include<bits/stdc++.h>
using namespace std;

const int N = 600;
const double INF = 1e9;

int n,m;

vector<int> adj[605];

double memo[605][5];

double dp(int x, int done){
	if(x == n) return 0.0;
	double &ret = memo[x][done];
	if(ret>=0.0) return ret;
	ret = 0;
	for(int i=0;i<adj[x].size();i++){
		int cur = adj[x][i];
		ret += dp(cur,done)+1;
	}
	double lol = ret/adj[x].size();
	ret = lol;
	if(done!=1 && adj[x].size() > 1){
		double lol2;
		double lala = 0.0;
		for(int i=0;i<adj[x].size();i++){
			int cur = adj[x][i];
			lala += dp(cur,1)+1;
		}
		lol2 = lala;
		for(int i=0;i<adj[x].size();i++){
			int cur = adj[x][i];
			double now = lala - dp(cur,1) - 1;
			lol2 = min(lol2,now);
		}
		lol2/=(adj[x].size()-1);
		ret = min(ret,lol2);
	}
	return ret;
}

int main(){
	for(int i=0;i<=601;i++){
		for(int j=0;j<3;j++){
			memo[i][j] = -1.0;
		}
	}
	scanf("%d%d",&n,&m);
	for(int i=1;i<=m;i++){
		int u,v;
		scanf("%d%d",&u,&v);
		adj[u].push_back(v);
	}
	printf("%.9lf\n",dp(1,0));
	return 0;
}