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 Python (3.4.3) (judge verdict: Wrong Answer)

N, M = map(int, input().split())
X = [0] * N
t = 0
for _ in range(M):
    a, b = map(int, input().split())
    X[a-1] += 1
    X[b-1] += 1
    t += a+b
    
for x in X:
    if x % 2 > 0:
        print("No")
        break
else:
    c = 0
    for x in X:
        if x == 4:
            c += 1
        if x >= 6:
            c = 3
            break
    if c >= 3:
        print("Yes")
    elif c == 2:
        if t % 50 in [6, 8, 10, 18, 36, 38] or (t%50) < 25 or (t%50) > 37:
            print("No")
        else:
            print("Yes")
    else:
        print("No")