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:
n,m = map(int,input().split())
ab = [list(map(int,input().split())) for i in range(m)]
go = [[] for i in range(n+1)]
come = [[] for i in range(n+1)]
for a,b in ab:
  go[a].append(b)
  come[b].append(a)
dpe = [0]*(n+1)
for i in range(1,n+1)[::-1]:
  if i < n and go[i]:
    dpe[i] /= len(go[i])
  for j in come[i]:
    dpe[j] += 1+dpe[i]
dpp = [0]*(n+1)
dpp[1] = 1
for i in range(1,n+1):
  for j in go[i]:
    dpp[j] += dpp[i]/len(go[i])
ans = dpe[1]
for a,b in ab:
  pa = dpp[a]
  if len(go[a]) == 1:
    continue
  ea = dpe[a]-1-((dpe[a]-1)*len(go[a])-dpe[b])/(len(go[a])-1)
  ans = min(ans,dpe[1]-pa*ea)

print(ans)