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 CPP):

Solution:
#include<iostream>
#include<iomanip>
#include<vector>
using namespace std;
int N,M;
vector<int>G[666];
double dp[666];
double x[666][666];
int main()
{
	cin>>N>>M;
	for(int i=0;i<M;i++)
	{
		int s,t;cin>>s>>t;
		s--,t--;
		G[s].push_back(t);
	}
	dp[N-1]=0;
	x[N-1][N-1]=1;
	for(int i=N-2;i>=0;i--)
	{
		double sum=0;
		for(int v:G[i])
		{
			sum+=dp[v];
			for(int j=0;j<N;j++)x[i][j]+=x[v][j]/G[i].size();
		}
		x[i][i]=1;
		dp[i]=sum/G[i].size()+1;
	}
	double ans=dp[0];
	for(int i=0;i<N-1;i++)
	{
		if(G[i].size()==1)continue;
		double sum=0,m=0;
		for(int v:G[i])
		{
			sum+=dp[v];
			m=max(m,dp[v]);
		}
		double ch=(sum-m)/(G[i].size()-1)+1-dp[i];
		ans=min(ans,dp[0]+ch*x[0][i]);
	}
	cout<<fixed<<setprecision(16)<<ans<<endl;
}