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 C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

//#pragma GCC optimize("Ofast,unroll-loops")
//#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,tune=native")
#include<bits/stdc++.h>

using namespace std;

#define all(x) (x).begin(),(x).end()
#define I(x, a) for(auto x : (a))
#define ins insert
#define F(i, l, r) for(auto i = (l); i < (r); i++)
#define DF(i, l, r) for(auto i = (l); i >= (r); i--)
#define E(i, l, r) for(auto i = (l); i <= (r); i++)
#define pb push_back
#define X first
#define Y second
#define mp make_pair
#define ret return
#define brk break
#define cont continue
#define move fjgjgjgjghd

typedef long long ll;
typedef unsigned long long ull;
typedef long double dbl;
typedef pair<int, int> pii;

const int N = (int)1e5 + 55;
multiset<int> gr[N];
int n, m;
bool was[N];

void fail(){
    cout << "No";
    exit(0);
}

int main(){
    ios_base::sync_with_stdio(false);
    cin.tie(0);
    int returned = 0;
    cin >> n >> m;
    F(i, 0, m){
        int v, u;
        cin >> v >> u;
        --v; --u;
        gr[v].insert(u);
        gr[u].insert(v);
    }
    F(i, 0, n)if(gr[i].size()&1)fail();
    stack<int> st;
    st.push(0);
    was[0] = 1;
    while(!st.empty()){
        int v = st.top();
        while(!gr[v].empty()){
            int u = *gr[v].begin();
            gr[v].erase(gr[v].begin());
            gr[u].erase(gr[u].find(v));
            st.push(u);
            if(was[u])++returned;
            was[u] = 1;
            brk;
        }
        if(v == st.top())st.pop();
    }
    if(returned < 3)fail();
    else cout << "Yes";
    return 0;
}