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>
#define REP(i,a,b) for(auto i=(a);i<(b);++i)
using namespace std;const int inf=1e9,N=600;bool adj[N][N];int deg[N];double E[N][2];signed main(){ios_base::sync_with_stdio(0);cin.tie(NULL);int n,m;cin>>n>>m;while(m--){int s,t;cin>>s>>t;--s,--t;adj[s][t]=1;++deg[s];}auto solve=[&](bool b){REP(k,1,n){int i=n-k-1;E[i][b]=0;REP(j,i+1,n-1)if(adj[i][j])E[i][b]+=E[j][b];E[i][b]/=deg[i];++E[i][b];}};solve(0);double ans=E[0][0];REP(i,0,n-1){if(deg[i]==1) continue;int mxpos;double mx=-1;REP(j,i+1,n)if(adj[i][j])if(mx<E[j][0]){mxpos=j;mx=E[j][0];}--deg[i];adj[i][mxpos]=0;solve(1);++deg[i];adj[i][mxpos]=1;ans=min(ans,E[0][1]);}cout<<setprecision(10)<<ans;}