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:
// In the name of God

#include <bits/stdc++.h>

using namespace std;

const int N = 605;

vector<int> go[N];
vector<pair<int,int>> edges;
int n, outdeg[N];

int main() {
	ios_base::sync_with_stdio(false);
	cin.tie(0);
	int m;
	cin >> n >> m;
	for(int i=0;i<m;i++) {
		int u, v;
		cin >> u >> v;
		u--;
		v--;
		edges.push_back(make_pair(u, v));
		go[u].push_back(v);
		outdeg[u]++;
	}
	vector<double> P(n, 0);
	P[0] = 1;
	for(int i=0;i<n-1;i++) {
		for(auto v : go[i]) P[v] += P[i] / outdeg[i];
	}
	vector<double> E(n, 0);
	for(int i=n-2;i>=0;i--) {
		for(auto v : go[i]) E[i] += E[v] / outdeg[i];
		E[i]++;
	}
	double ans = E[0];
	for(int i=0;i<m;i++) {
		int from = edges[i].first, to = edges[i].second;
		if(outdeg[from] == 1) continue;
		double en = E[from] - 1;
		en = en * outdeg[from] - E[to];
		en = en / (outdeg[from] - 1) + 1;
		ans = min(ans, (en-E[from])*P[from] + E[0]);
	}
	cout << setprecision(10) << fixed << ans;
	return 0;
}