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:
def solve_f(n, g):
    e_base = [0.0] * (n + 1)
    p_base = [0.0] * (n + 1)

    # e_base
    for i in range(n - 1, 0, -1):
        e_base[i] = sum([e_base[j] for j in g[i]]) / len(g[i]) + 1.0

    # p_base
    p_base[1] = 1.0
    for i in range(1, n):
        for j in g[i]:
            p_base[j] += p_base[i] / len(g[i])

    # print(p_base, e_base)

    # find maximal gap
    gap = 0.0
    for i in range(1, n):
        if len(g[i]) == 1:
            continue
        total = sum([e_base[j] + 1 for j in g[i]])
        maxim = max([e_base[j] + 1 for j in g[i]])
        gap_i = (total / len(g[i]) - (total - maxim) / (len(g[i]) - 1)) * p_base[i]
        gap = max(gap_i, gap)

    return e_base[1] - gap


def main():
    # input
    n, m = map(int, input().split())
    g = [[] for _ in range(n)]
    for _ in range(m):
        s, t = map(int, input().split())
        g[s].append(t)
    res = solve_f(n, g)
    print(res)


def test():
    assert abs(solve_f(4, [[], [2, 3, 4], [3, 4], [4]]) - 1.5) < 0.000001
    assert abs(solve_f(3, [[], [2], [3]]) - 2.0) < 0.000001


if __name__ == "__main__":
    test()
    main()