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### forkintheroad

### 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`) and can only be traversed in the direction from Room

_{i}`s`to Room

_{i}`t`. It is known that, for each room except Room

_{i}`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 ≤ N ≤ 600``N-1 ≤ M ≤ N(N-1)/2``s`_{i}< t_{i}- If
`i != j`,`(s`._{i}, t_{i}) ≠ (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:

NMs_{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

### Tags

### Subtasks and Limits

Subtask | Score | #TC | Time | Memory | Scoring |
---|---|---|---|---|---|

1 | 100 | 65 | 2s | 256MB | Minimum |

2 | 0 | 3 | 2s | 256MB | Minimum |

### Judge Compile Command

g++-8 ans.cpp -o forkintheroad -Wall -Wshadow -static -O2 -lm -m64 -s -w -std=gnu++17 -fmax-errors=512