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
import copy

def main():
	input = sys.stdin.readline

	N, M = list(map(int,input().split()))
	st = [list(map(int,input().split())) for i in range(M)]

	neighbor = [[] for i in range(N)]
	for s,t in st:
		neighbor[s-1].append(t-1)

	INF = 100100100
	ev = [INF]*N
	ev[N-1] = 0
	for i in range(N-2,-1,-1):
		dsum = 0
		dcount = 0
		for j in neighbor[i]:
			dsum += ev[j]
			dcount += 1
		ev[i] = dsum/dcount + 1

	ans = ev[0]
	for i in range(N-1):
		evmax = -1
		if len(neighbor[i])==1: continue
		for j in neighbor[i]:
			if ev[j]>evmax:
				tn = j
				evmax = ev[j]

		# ntmp = copy.deepcopy(neighbor)
		# ntmp[i].remove(tn)
		evtmp = [INF]*N
		evtmp[i+1::] = ev[i+1::]
		for ii in range(i,-1,-1):
			dsum = 0
			dcount = 0
			for jj in neighbor[ii]:
				if ii==i and jj==tn: continue
				dsum += evtmp[jj]
				dcount += 1
			evtmp[ii] = dsum/dcount + 1

		if evtmp[0]<ans:
			ans = evtmp[0]

	print(ans)

if __name__ == '__main__':
    main()