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 <vector>
#include <iomanip>
using namespace std;

int main(){
    int N,M;
    cin >> N >> M;
    vector<vector<int>> g(N);
    vector<int> indeg(N);
    for(int i=0;i<M;i++){
        int a,b;
        cin >> a >> b;
        a--; b--;
        g[b].push_back(a);
        indeg[a]++;
    }
    auto calc = [&](int id){
        vector<double> dp(N,0);
        vector<vector<double>> dplist(N);
        for(int i=N-1;i>=0;i--){
            double sum = 0,ma = 0;
            for(auto& x:dplist[i]){
                sum += x;
                ma = max(ma,x);
            }
            if(id==i) dp[i] = (sum-ma)/(indeg[i]-1);
            else dp[i] = (i!=N-1? sum/indeg[i]:0);
            for(auto& e:g[i]){
                dplist[e].push_back(1+dp[i]);
            }
        }
        return dp[0];
    };
    double ans = 1e18;
    for(int i=0;i<=N;i++) if(indeg[i]!=1) ans = min(ans,calc(i));
    cout << fixed << setprecision(10) << ans << endl;
}