Problem:
Score : 600 points

Problem Statement

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}.

Sample Input 1

4 6
1 4
2 3
1 3
1 2
3 4
2 4

Sample Output 1

1.5000000000

If Aoki blocks the passage from Room 1 to Room 2, Takahashi will go along the path 1 → 3 → 4 with probability \frac{1}{2} and 1 → 4 with probability \frac{1}{2}. E = 1.5 here, and this is the minimum possible value of E.

Sample Input 2

3 2
1 2
2 3

Sample Output 2

2.0000000000

Blocking any one passage makes Takahashi unable to reach Room N, so Aoki cannot block a passage.

Sample Input 3

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

Sample Output 3

3.0133333333

Solution in PyPy3 (2.4.0) (judge verdict: Time Limit Exceeded)

import sys
import copy

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

INF = 100100100
ev = [INF]*N
ev[N-1] = 0
for i in range(N-2,-1,-1):
	dlist = []
	for j in neighbor[i]:
		dlist.append(ev[j])
	ev[i] = sum(dlist)/len(dlist) + 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[N-1] = 0
	for ii in range(N-2,-1,-1):
		dlist = []
		for jj in ntmp[ii]:
			dlist.append(evtmp[jj])
		evtmp[ii] = sum(dlist)/len(dlist) + 1

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

print(ans)