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<bits/stdc++.h>
#define maxn 605
#define maxm maxn*maxn/2
using namespace std;
int n,m,deg[maxn],stk[maxn],top;
double f[maxn],g[maxn],ans;
vector<int>G[maxn];
double dfs(int u){
	if(u==n) return 0;
	if(f[u]) return f[u];
	for(int v: G[u]) f[u]+=dfs(v);
	return f[u]=f[u]/G[u].size()+1;
}
int main()
{
	scanf("%d%d",&n,&m);
	for(int i=1,x,y;i<=m;i++) scanf("%d%d",&x,&y),G[x].push_back(y),deg[y]++;
	dfs(1);
	g[1]=1;
	for(int i=1;i<=n;i++) if(!deg[i]) stk[++top]=i; 
	while(top){
		int u=stk[top--]; if(u==n) break;
		for(int v: G[u]){
			g[v]+=g[u]/G[u].size();
			if(!--deg[v]) stk[++top]=v;
		}
	}
	for(int u=1;u<n;u++) if(G[u].size()>1){
		double tmp=0; for(int v: G[u]) tmp+=f[v];
		for(int v: G[u])
			ans=max(ans,g[u]*(f[u]-((tmp-f[v])/(G[u].size()-1)+1)));
	}
	printf("%.10f\n",f[1]-ans);
}