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 <cmath>
#include <vector>
#include <iomanip>

int main()
{
    std::cout << std::setprecision(10);
    int N, M;
    std::cin >> N >> M;

    std::vector<std::vector<int>> edge(N);
    for (int i=0; i<M; ++i) {
        int s, t;
        std::cin >> s >> t;
        edge[s-1].push_back(t-1);
    }

    double opt = N+M;
    for (int i=0; i<N-1; ++i) { // 削除する部屋でループ
        std::vector<double> v(N);
        v[N-1] = 0.0;
        for (int j=N-2; j>=0; --j) { // DP計算
            double sum = 0.0;
            double maxv = 0.0;
            for (int t : edge[j]) { // 部屋内通路ループ
                sum += v[t];
                maxv = std::max(maxv, v[t]);
            }
            int nLink = edge[j].size();
            if (nLink > 1 && i == j) {
                sum -= maxv;
                nLink -= 1;
            }
            v[j] = (sum + nLink) / nLink;
        }
        opt = std::min(opt, v[0]);
    }

    std::cout << opt << std::endl;
    return 0;
}