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;

typedef long long LL;
const int N = 604;
const double INF = 1e100;
vector<int> out[N];
double dp[N];

int main() {
    ios::sync_with_stdio(false);
    cin.tie(0);


    int n, m;
    cin>>n>>m;

    for (int i=1; i<=m; i++) {
        int u, v;
        cin>>u>>v;
        out[u].push_back(v);
    }

    dp[n] = 0;
    double ans = INF;
    for (int i=n-1; i>0; i--) {
        double sum = 0;
        int id = 0;

        for (int x: out[i]) {
            if (dp[x] > dp[id])
                id = x;
            sum += dp[x];
        }
        int k = out[i].size();

        dp[i] = k > 1 ? 1 + (sum-dp[id])/(k-1) : INF;

        for (int j=i-1; j>0; j--) {
            double sum = 0;
            int k = out[j].size();
            for (int x: out[j])
                sum += dp[x];
            dp[j] = 1 + sum/k;
        }
        ans = min(ans, dp[1]);
        dp[i] = 1 + sum/k;
    }
    ans = min(ans, dp[1]);
    cout<<setprecision(10)<<fixed<<ans<<endl;
}