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:

from collections import defaultdict,deque
import sys,heapq,bisect,math,itertools,string,queue,copy,time
sys.setrecursionlimit(10**8)
INF = float('inf')
mod = 10**9+7
eps = 10**-7
def inp(): return int(sys.stdin.readline())
def inpl(): return list(map(int, sys.stdin.readline().split()))
def inpl_str(): return list(sys.stdin.readline().split())

N,M = inpl()

if M == N-1: # 一本道
    for _ in range(M):
        s,t = inpl()
    print(M)
else:
    lines = defaultdict(set)
    LL = [0]*N
    for _ in range(M):
        s,t = inpl()
        s -= 1; t -= 1
        lines[s].add(t)
        LL[s] += 1

    def dfs(s):
        global MEMO
        if MEMO[s] != -1:
            return MEMO[s]
        else:
            tmp = 0
            for t in lines[s]:
                if MEMO[t] != -1:
                    tmp += MEMO[t] + 1
                else:
                    tmp += dfs(t) + 1
            MEMO[s] = tmp / LL[s]
            return MEMO[s]

    MEMO = [-1]*N
    MEMO[-1] = 0
    dfs(0)
    koho = [-1]*N
    for s in range(N):
        x = -1
        for t in lines[s]:
            if x < MEMO[t]:
                x = MEMO[t]
                koho[s] = t

    ans = 10**10
    for s in range(N-1):
        t = koho[s]
        if LL[s] == 1 or t == -1:
            continue
        MEMO = [-1]*N
        MEMO[-1] = 0
        lines[s].remove(t)
        LL[s] -= 1
        ans = min(ans,dfs(0))
        lines[s].add(t)
        LL[s] += 1
    print(ans)