Contest Duration: ~ (local time) (100 minutes) Back to Home
F - Fork in the Road /

Time Limit: 2 sec / Memory Limit: 1024 MB

### 問題文

N 個の部屋と、M 本の一方向にのみ通れる通路から成る洞窟があります。部屋には 1 から N までの番号がついています。

### 制約

• 2 ≤ N ≤ 600
• N-1 ≤ M ≤ \frac{N(N-1)}{2}
• s_i < t_i
• i \neq j のとき、(s_i, t_i) \neq (s_j, t_j) (21:23 追記)
• 任意の v = 1, 2, ..., N-1 に対し、ある i が存在して v = s_i

### 入力

N M
s_1 t_1
:
s_M t_M


### 入力例 1

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


### 出力例 1

1.5000000000


### 入力例 2

3 2
1 2
2 3


### 出力例 2

2.0000000000


どの通路を塞いでも部屋 N に到達出来なくなるため、青木君は通路を塞ぐことは出来ません。

### 入力例 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


### 出力例 3

3.0133333333


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 134 with probability \frac{1}{2} and 14 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