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 resolve():
    n,m=map(int,input().split())
    edges=[[] for _ in range(n)]
    for _ in range(m):
        s,t=map(int,input().split())
        s-=1; t-=1
        edges[s].append(t)
 
    # 確率DP
    p=[0]*n
    p[0]=1.0
    for v in range(n):
        for nv in edges[v]:
            p[nv]+=p[v]/len(edges[v])
 
    # 期待値DP
    E=[0]*n
    for v in range(n-2,-1,-1):
        for nv in edges[v]:
            E[v]+=E[nv]
        E[v]=1+E[v]/len(edges[v])
 
    ans=E[0]
    for v in range(n):
        if(len(edges[v])<=1): continue
        # 各vに対して、辺vwのうちE[w]が最大のものを削除する
        w=max(*((E[w],w) for w in edges[v]))[1]
        dE=0
        deg=len(edges[v])
        for nv in edges[v]:
            if(nv==w): dE-=E[nv]/deg
            else: dE+=E[nv]/(deg-1)/deg
        ans=min(ans,E[0]+p[v]*dE)
 
    print(ans)
resolve()