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 JAVA):

Solution:
import java.util.*;

public class Main implements Runnable {

    private static int MOD = 1_000_000_007;

    public static void main(String[] args) {
        // Run with 32MB stack
        Thread thread = new Thread(null, new Main(), "", 32 * 1024 * 1024);
        thread.start();
    }

    @Override
    public void run() {
        final Scanner scanner = new Scanner(System.in);
        solve(scanner);
    }

    static void solve(Scanner scanner) {
        int N = Integer.parseInt(scanner.next());
        int M = Integer.parseInt(scanner.next());
        List<Integer>[] edges = new List[N];
        for (int i = 0; i < N; i++) {
            edges[i] = new ArrayList<>();
        }
        for (int i = 0; i < M; i++) {
            int s = Integer.parseInt(scanner.next());
            int t = Integer.parseInt(scanner.next());
            edges[s - 1].add(t - 1);
        }

        double[] dp = new double[N];
        for(int i = N - 2; i >= 0; i--) {
            double sum = 0;
            for (int next: edges[i]) {
                sum += dp[next];
            }
            sum /= edges[i].size();
            sum += 1;
            dp[i] = sum;
        }

        double ans = dp[0];
        for (int ni = 0; ni < N - 1; ni++) {
            if (edges[ni].size() == 1) continue;
            int nj = edges[ni].get(0);
            for (int next: edges[ni]) {
                if (dp[nj] < dp[next]) {
                    nj = next;
                }
            }
            // ignore edge (ni -> nj)

            double[] dp2 = new double[N];
            for(int i = N - 2; i >= 0; i--) {
                double sum = 0;
                for (int next: edges[i]) {
                    if (i == ni && next == nj) continue;
                    sum += dp2[next];
                }
                int deg = edges[i].size();
                if (i == ni) deg--;
                sum /= deg;
                sum += 1;
                dp2[i] = sum;
            }

            ans = Math.min(ans, dp2[0]);
        }

        System.out.printf("%.10f\n", ans);
    }
}