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;

double dp[605][605], e[605];
vector<int> g[605], nx[605];

int main() {
	ios::sync_with_stdio(false); cin.tie(0); cout.tie(0);
	int n, m;
	cin >> n >> m;
	for (int i = 1; i <= m; i++) {
		int u, v;
		cin >> u >> v;
		g[v].push_back(u);
		nx[u].push_back(v);
	}
	e[n] = 0;
	for (int i = n - 1; i >= 1; i--) {
		e[i] = 1;
		for (auto v : nx[i]) {
			e[i] += e[v] / nx[i].size();
		}
	}
	for (int i = n - 1; i >= 1; i--) {	
		for (auto v : nx[i]) {
			for (int j = i + 1; j <= n; j++) {
				dp[i][j] += dp[v][j] / nx[i].size();
			}
		}
		dp[i][i] = 1;
	}
	double mn = 0;
	for (int i = 1; i <= n; i++) {
		if (nx[i].size() < 2) continue;
		for (auto v : nx[i]) {
			double minus = (e[i] - 1) * nx[i].size();
			minus -= e[v];
			//if (i == 1 && v == 2) cout << minus << endl;
			minus = minus / (nx[i].size() - 1) + 1 - e[i];
			//if (i == 1 && v == 2) cout << minus << endl;
			minus = minus * dp[1][i];
			//if (i == 1 && v == 2) cout << minus << endl;
			mn = min(mn, minus);
			
		}
	}
	cout << setprecision(10) << fixed << e[1] + mn;
	return 0;
}