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<cstdio>
#include<algorithm>
#include<cstring>
#include<cmath>
#include<vector>
using namespace std;
typedef long long ll;
typedef double db;
inline int read()
{
	int x=0,f=1; char ch=getchar();
	while(ch<'0'||ch>'9') { if(ch=='-') f=-1; ch=getchar(); }
	while(ch>='0'&&ch<='9') { x=(x<<1)+(x<<3)+(ch^48); ch=getchar(); }
	return x*f;
}
const int N=607,INF=1e9;
int n,m,a[N*N],b[N*N];
vector <int> V[N];
db f[N],ans;
inline db calc(int p)
{
	for(int i=1;i<=n;i++) f[i]=0;
	f[n]=0;
	for(int i=n-1;i>=1;i--)
	{
		int len=V[i].size();
		if(p==i&&len==1) { f[i]=INF; continue; }
		db t=1.0/(len-(p==i)),mx=0;
		for(int j=0;j<len;j++)
		{
			int v=V[i][j]; mx=max(mx,(f[v]+1)*t);
			f[i]+=(f[v]+1)*t;
		}
		if(p==i) f[i]-=mx;
	}
	return f[1];
}
int main()
{
	n=read(),m=read();
	for(int i=1;i<=m;i++)
	{
		a[i]=read(),b[i]=read();
		V[a[i]].push_back(b[i]);
	}
	ans=calc(0);
	for(int i=1;i<=n;i++)
		ans=min(ans,calc(i));
	printf("%.9lf\n",ans);
	return 0;
}