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 PYTHON3):

Solution:
#!/usr/bin/env python3
import sys
from collections import Counter
INF = float("inf")


class Graph(object):
    def __init__(self, N):
        self.N = N
        self.E = [[] for _ in range(N)]

    def add_edge(self, s, t, w=1):
        self.E[s].append(t)


def solve(N: int, M: int, s: "List[int]", t: "List[int]"):
    c = Counter(s)
    g = Graph(N)
    for ss, tt in zip(s, t):
        g.add_edge(ss, tt)

    DP = [0]*N
    for i in range(N-2, -1, -1):
        DP[i] = sum(DP[t] for t in g.E[i])/c[i] + 1

    gm = DP[0]
    for i in range(N-2):
        if len(g.E[i]) == 1:
            continue
        dp = DP[:]
        for j in range(i, -1, -1):
            tot, m = 0, 0
            for t in g.E[j]:
                tot += dp[t]
                m = max(dp[t], m)
            if i == j:
                dp[j] = (tot-m)/(c[j]-1)
            else:
                dp[j] = tot/c[j]
            dp[j] += 1
        gm = min(dp[0], gm)
        # print(dp)
    print(gm)


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    M = int(next(tokens))  # type: int
    s = [int()] * (M)  # type: "List[int]"
    t = [int()] * (M)  # type: "List[int]"
    for i in range(M):
        s[i] = int(next(tokens))-1
        t[i] = int(next(tokens))-1
    solve(N, M, s, t)


if __name__ == '__main__':
    main()