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;
using Graph = vector<set<int>>;
 
int main() {
  int N, M;
  cin >> N >> M;
  Graph E(N);
  int a, b;
  for (int i=0; i < M; i++) {
    cin >> a >> b; a--; b--;
    E.at(a).insert(b);
  }
  vector<double> ex(N, 0); double ans;
  vector<int> mx(N-1, 0);
  ex.at(N - 1) = 0;
  for (int i=N-2; i >= 0; i--) {
    double tmp = -1;
    for (int x : E.at(i)) {
      ex.at(i) += ex.at(x);
      if (ex.at(x) > tmp) {
        mx.at(i) = x;
        tmp = ex.at(x);
      }
    }
    ex.at(i) = ex.at(i) / E.at(i).size() + 1;
  }
  ans = 0;
  vector<double> p(N, 0); p.at(0) = 1;
  for (int i=0; i < N; i++) {
    for (int x : E.at(i)) {
      p.at(x) += 1.0 * p.at(i) / E.at(i).size();
    }
  }
  for (int i=N-2; i >= 0; i--) {
    double ex2 = 0;
    if (E.at(i).size() > 1) {
      for (int x : E.at(i)) {
        ex2 += ex.at(x);
      }
      ex2 = ( ex2 - ex.at(mx.at(i)) ) / ( E.at(i).size() - 1) + 1;
      ans = max(ans, p.at(i) * (ex.at(i) - ex2));
    }
  }
  printf("%.8lf\n", ex.at(0) - ans);
}