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
sys.setrecursionlimit(10**7)
inf=float('inf')

def main(n,m,g,uv):
    # 1<=E<=n-1
    eary=[inf]*m
    vary=[inf]*n
    vary[-1]=0
    todo=[n-1]
    def dfs(v,x):
        if vary[v]<inf:return vary[v]
        ret=0
        cnt=0
        for i,nv in g[v]:
            if i==x:continue
            tmp=dfs(nv,x)+1
            eary[i]=tmp
            ret+=tmp
            cnt+=1
        if cnt==0:
            for i in range(n):
                vary[i]=inf
            return inf
        vary[v]=ret/cnt
        return ret/cnt
    e0=dfs(0,-1)
    ary=[-1]*m
    bary=[0]*n
    bary[0]=1
    for v in range(n):
        k=len(g[v])
        e=vary[v]
        for i,nv in g[v]:
            bary[nv]+=1/k*bary[v]
            if k>1 and bary[v]:
                ary[i]=max(0,e-(e*k-eary[i])/(k-1))
                ary[i]*=bary[v]
    return e0-max(0,max(ary))

n,m=map(int,input().split())
uv=[list(map(int,input().split())) for _ in range(m)]
g=[[] for _ in range(n)]
for i,(u,v) in enumerate(uv):
    u,v=u-1,v-1
    g[u].append((i,v))
print(main(n,m,g,uv))