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 ld = long double ;
int n, m;
vector<int> a[605];
ld dp[605];

ld calc (int x) {
    dp[n] = 0 ;
    for (int cur = n - 1 ; cur >= 1; -- cur) {
        ld sum = 0.0;
        ld ma = numeric_limits<ld>::min();
        for (int j : a[cur]) {
            sum += dp[j];
            ma = max(ma, dp[j]);
        }
        if (cur == x && (int)a[cur].size() >= 2) {
            sum -= ma;
            sum /= (ld) (a[cur].size() - 1);
        }
        else sum /= (ld) (a[cur].size());
        dp[cur] = sum + 1;
    }
    return dp[1];

}

int main () {
    scanf("%d %d", &n, &m);
    for (int i = 1 ; i <= m ; ++ i) {
        int from , to ;
        scanf("%d %d", &from, &to);
        a[from].emplace_back(to);
    }
    ld ans = numeric_limits<ld>::max();
    for (int i = 0 ; i < n ; ++ i) {
        ans = min(ans, calc(i));
    }
    cout << fixed << setprecision(10) << ans << '\n';
}