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>
using namespace std;
int main(){
    int n,m;
    cin>>n>>m;
    int s[m],t[m];
    vector<vector<int>>e(n);
    for(int i=0;i<m;i++){
        cin>>s[i]>>t[i];
        e[s[i]-1].push_back(t[i]-1);
    }
    vector<double>dp(n);
    vector<double>ma(n);
    dp[n-1]=0.0;
    for(int i=n-2;i>=0;i--){
        double x=0.0;
        for(int j=0;j<e[i].size();j++){
            x+=dp[e[i][j]];
            ma[i]=max(ma[i],dp[e[i][j]]);
        }
        dp[i]=x/double(e[i].size())+1.0;
    }
    double ans=dp[0];
    for(int i=0;i<n-2;i++){
        if(e[i].size()==1)continue;
        vector<double>dp_change(n);
        dp_change=dp;
        for(int j=i;j>=0;j--){
            double x=0.0;
            for(int k=0;k<e[j].size();k++)x+=dp_change[e[j][k]];
            if(j==i)dp_change[j]=(x-ma[j])/double(e[j].size()-1)+1.0;
            else dp_change[j]=x/double(e[j].size())+1.0;
        }
        ans=min(ans,dp_change[0]);
    }
    printf("%.10lf\n",ans);
    return 0;
}