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 Rust (1.42.0) (judge verdict: Accepted)

use std::io::*;
use std::str::FromStr;

#[allow(dead_code)]
fn read<T: FromStr>() -> T {
    let stdin = stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("failed to read char") as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().expect("failed to parse token")
}

fn calc(
    target: usize,
    depth: usize,
    nexts: &Vec<Vec<usize>>,
    memo: &mut Vec<Vec<Option<f64>>>,
) -> f64 {
    let n = nexts.len();
    let target = if target < depth { 0 } else { target };
    if depth == n - 1 {
        return 0.0;
    }
    if let Some(v) = memo[target][depth] {
        return v;
    }
    let mut nrets = vec![];
    let mut sum = 0.0;
    for &t in nexts[depth].iter() {
        let v = 1.0 + calc(target, t, nexts, memo);
        sum += v;
        nrets.push(v);
    }
    let mut ret = sum / nexts[depth].len() as f64;
    if target == depth && nexts[depth].len() != 1 {
        nrets.sort_by(|lhs, rhs| lhs.partial_cmp(rhs).unwrap());
        sum -= nrets.last().unwrap();
        ret = sum / (nexts[depth].len() - 1) as f64;
    }
    memo[target][depth] = Some(ret);
    return ret;
}

fn main() {
    let n = read::<usize>();
    let m = read::<usize>();
    let mut nexts = vec![vec![]; n];
    for _i in 0..m {
        let s = read::<usize>() - 1;
        let t = read::<usize>() - 1;
        nexts[s].push(t);
    }
    let mut memo = vec![vec![None; n]; n];
    let mut ans = 1e+10;
    for i in 0..n {
        let nans = calc(i, 0, &nexts, &mut memo);
        if nans < ans {
            ans = nans;
        }
    }
    println!("{:.8}", ans)
}