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())
R=[tuple(map(int,input().split())) for i in range(M)]

E=[set() for i in range(N)]

for x,y in R:
    E[x-1].add(y-1)

DP=[M]*N
DP[N-1]=0

for i in range(N-2,-1,-1):
    score=0
    for x in E[i]:
        score+=1+DP[x]

    DP[i]=score/len(E[i])
        
ANS=DP[0]

for i in range(N-2):
    rem=-1
    remsc=0
    for j in E[i]:
        if DP[j]>remsc:
            rem=j
            remsc=DP[j]

    if remsc==0:
        continue

    E[i].remove(rem)

    NDP=[M]*N
    NDP[N-1]=0

    for l in range(N-2,-1,-1):
        
        score=0
        
        if len(E[l])==0:
            continue
        
        for x in E[l]:
            score+=1+NDP[x]

        NDP[l]=score/len(E[l])

    #print(i,rem,E,NDP)

    ANS=min(ANS,NDP[0])
    E[i].add(rem)

print(ANS)