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 <algorithm>
#include <vector>
using namespace std;
#define N 605

double dp[N][2];
vector<int> adj[N];
double getMaxi(int block,int n) {
	dp[n][0] = 0, dp[n][1] = 1e18;
	for (int i = n - 1; i; --i) {
		double sum = 0, minZero = 1e18, minDiff = 1e18;
		int sz = adj[i].size();
		for (int j = 0; j < sz; ++j) {
			int id = adj[i][j];
			double zero = dp[id][0], one = dp[id][1];
			minDiff = min(minDiff, one - zero);
			minZero = min(minZero, -zero);
			sum += zero;
		}
		dp[i][0] = 1 + sum / sz;
		if (sz >= 2 && block == i) {
			dp[i][0] = 1 + (sum + minZero) / (sz - 1);
		}
	}
	return dp[1][0];
}
int main(void) {
	//solve 
	int n, m; scanf("%d %d", &n, &m);
	for (int i = 0; i < m; ++i) {
		int s, t; scanf("%d %d", &s, &t);
		adj[s].push_back(t);
	}
	double ret = 1e18;
	for (int i = n - 1; i; --i) {
		ret = min(ret, getMaxi(i, n));
	}
	printf("%0.9lf\n", ret);
	return 0;
}