Problem:
Score : 800 points

Problem Statement

You are given a simple connected undirected graph consisting of N vertices and M edges.
The vertices are numbered 1 to N, and the edges are numbered 1 to M.

Edge i connects Vertex a_i and b_i bidirectionally.

Determine if three circuits (see Notes) can be formed using each of the edges exactly once.

Notes

A circuit is a cycle allowing repetitions of vertices but not edges.

Constraints

All values in input are integers.

1 \leq N,M \leq 10^{5}

1 \leq a_i, b_i \leq N

The given graph is simple and connected.

Input

Input is given from Standard Input in the following format:

N M
a_1 b_1
:
a_M b_M

Output

If three circuits can be formed using each of the edges exactly once, print Yes; if they cannot, print No.

Sample Input 1

7 9
1 2
1 3
2 3
1 4
1 5
4 5
1 6
1 7
6 7

Sample Output 1

Yes

Three circuits can be formed using each of the edges exactly once, as follows:

Sample Input 2

3 3
1 2
2 3
3 1

Sample Output 2

No

Three circuits are needed.

Sample Input 3

18 27
17 7
12 15
18 17
13 18
13 6
5 7
7 1
14 5
15 11
7 6
1 9
5 4
18 16
4 6
7 2
7 11
6 3
12 14
5 2
10 5
7 8
10 15
3 15
9 8
7 15
5 16
18 15

Sample Output 3

Yes

Solution in Rust (1.15.1) (judge verdict: Wrong Answer)

use std::io::Read;

fn solve(g: &Vec<Vec<usize>>) -> bool {
    for v in g {
        if v.len()%2 == 1 { return false }
    }
    let mut p4 = Vec::new();
    for i in 0..g.len() {
        if g[i].len() >= 6 { return true }
        if g[i].len() == 4 { p4.push(i); }
    }
    if p4.len() != 1 { return p4.len() >= 3 }
    for &t in &g[p4[0]] {
        let mut prev = p4[0];
        let mut p = t;
        while g[p].len() == 2 {
            let next = g[p][if g[p][0] != prev { 0 } else { 1 }];
            prev = p;
            p = next;
        }
        if p == p4[0] { return true }
    }
    false
}

fn main() {
    let mut buf = String::new();
    std::io::stdin().read_to_string(&mut buf).ok();
    let mut it = buf.split_whitespace();
    let n = it.next().unwrap().parse::<usize>().unwrap();
    let m = it.next().unwrap().parse::<usize>().unwrap();
    let mut g = vec![Vec::new(); n];
    for _ in 0..m {
        let a = it.next().unwrap().parse::<usize>().unwrap() - 1;
        let b = it.next().unwrap().parse::<usize>().unwrap() - 1;
        g[a].push(b);
        g[b].push(a);
    }
    println!("{}", if solve(&g) { "Yes" } else { "No" });
}