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:
import sys
input = sys.stdin.readline


n, m = map(int, input().split())
edges_from = [[] for _ in range(n)]
edges_to = [[] for _ in range(n)]

for _ in range(m):
    _from, to = map(lambda x: int(x)-1, input().split())
    edges_from[_from].append(to)

E_dist_to_goal = [0] * n
P_of_reaching = [0] * n
P_of_reaching[0] = 1

for v in range(n-2, -1, -1):
    routes = edges_from[v]
    for next_v in routes:
        E_dist_to_goal[v] += E_dist_to_goal[next_v] + 1
    E_dist_to_goal[v] /= len(routes)

for v in range(n-1):
    P_now = P_of_reaching[v]
    routes = edges_from[v]
    P_next = P_now / len(routes)
    for next_v in routes:
        P_of_reaching[next_v] += P_next

initial_ans = E_dist_to_goal[0]
short_cut = 0
for v in range(n-2):
    routes = edges_from[v]
    num_of_routes = len(routes)
    if num_of_routes == 1:
        continue
    not_cut_dist = E_dist_to_goal[v]
    longest_dist = max([E_dist_to_goal[next_v]+1 for next_v in routes])
    after_cut_dist = (not_cut_dist - longest_dist/num_of_routes) * num_of_routes / (num_of_routes-1)
    delta = (not_cut_dist - after_cut_dist) * P_of_reaching[v]
    short_cut = max(short_cut, delta)

ans = initial_ans - short_cut
print(ans)