Problem:
Score : 700 points

Problem Statement

Snuke is playing a puzzle game.
In this game, you are given a rectangular board of dimensions R × C, filled with numbers. Each integer i from 1 through N is written twice, at the coordinates (x_{i,1},y_{i,1}) and (x_{i,2},y_{i,2}).

The objective is to draw a curve connecting the pair of points where the same integer is written, for every integer from 1 through N.
Here, the curves may not go outside the board or cross each other.

Determine whether this is possible.

Constraints

1 ≤ R,C ≤ 10^8

1 ≤ N ≤ 10^5

0 ≤ x_{i,1},x_{i,2} ≤ R(1 ≤ i ≤ N)

0 ≤ y_{i,1},y_{i,2} ≤ C(1 ≤ i ≤ N)

All given points are distinct.

All input values are integers.

Input

Input is given from Standard Input in the following format:

R C N
x_{1,1} y_{1,1} x_{1,2} y_{1,2}
:
x_{N,1} y_{N,1} x_{N,2} y_{N,2}

Output

Print YES if the objective is achievable; print NO otherwise.

Sample Input 1

4 2 3
0 1 3 1
1 1 4 1
2 0 2 2

Sample Output 1

YES

The above figure shows a possible solution.

Sample Input 2

2 2 4
0 0 2 2
2 0 0 1
0 2 1 2
1 1 2 1

Sample Output 2

NO

Sample Input 3

5 5 7
0 0 2 4
2 3 4 5
3 5 5 2
5 5 5 4
0 3 5 1
2 2 4 4
0 5 4 1

Sample Output 3

YES

Sample Input 4

1 1 2
0 0 1 1
1 0 0 1

Sample Output 4

NO

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include "iostream"
#include "climits"
#include "list"
#include "queue"
#include "stack"
#include "set"
#include "functional"
#include "algorithm"
#include "string"
#include "map"
#include "iomanip"
#include "random"

using namespace std;

const long long int MOD = 1000000007;
const long double EPS = 0.00000001;
const long double PI = 3.1415926535897932384626433;

long long int N, M, K, H, W, L, R;

int main() {
	ios::sync_with_stdio(false);
	cin.tie(0);

	cin >> H >> W >> N;
	vector<pair<int, int>>box;
	for (int i = 0; i < N; i++) {
		int sx, sy, gx, gy;
		cin >> sy >> sx >> gy >> gx;
		if (sx != 0 && sx != W&&sy!=0&&sy!=H) {
			continue;
		}
		if (gx != 0 && gx != W&&gy!=0&&gy!=H) {
			continue;
		}
		if (sy == 0) {
			box.push_back({ sx,i });
		}
		else if (sx == W) {
			box.push_back({ W + sy,i });
		}
		else if (sy == H) {
			box.push_back({ 2 * W + H - sx,i });
		}
		else {
			box.push_back({ 2 * W + 2 * H - sy,i });
		}
		if (gy == 0) {
			box.push_back({ gx,i });
		}
		else if (gx == W) {
			box.push_back({ W + gy,i });
		}
		else if (gy == H) {
			box.push_back({ 2 * W + H - gx,i });
		}
		else {
			box.push_back({ 2 * W + 2 * H - gy,i });
		}
	}
	if (box.empty()) {
		cout << "YES\n";
		return 0;
	}
	sort(box.begin(), box.end());
	M = box.size();
	if (M <= 2) {
		cout << "YES\n";
		return 0;
	}
	queue<int>Q;
	for (int i = 0; i < M; i++) {
		if (box[i].second == box[(i + 1) % M].second) {
			Q.push(i);
		}
	}
	if (Q.empty()) {
		cout << "NO\n";
		return 0;
	}
	vector<int>nx(M);
	vector<int>bf(M);
	vector<bool>flag(M, true);
	for (int i = 0; i < M; i++) {
		nx[i] = (i + 1) % M;
		bf[i] = (i - 1 + M) % M;
	}
	K = M;
	int l = Q.front();
	Q.pop();
	int r = l + 1;
	r %= M;
	while (K > 2) {
		if (box[l].second == box[r].second) {
			flag[l] = false;
			flag[r] = false;
			K -= 2;
			bf[nx[r]] = bf[l];
			nx[bf[l]] = nx[r];
			l = bf[l];
			r = nx[r];
			continue;
		}
		if (box[l].second == box[bf[l]].second) {
			flag[l] = false;
			flag[bf[l]] = false;
			K -= 2;
			bf[nx[l]] = bf[bf[l]];
			nx[bf[bf[l]]] = nx[l];
			l = bf[bf[l]];
			continue;
		}
		if (box[r].second == box[nx[r]].second) {
			flag[r] = false;
			flag[nx[r]] = false;
			K -= 2;
			bf[nx[nx[r]]] = bf[r];
			nx[bf[r]] = nx[nx[nx[r]]];
			r = nx[nx[r]];
			continue;
		}
		if (Q.empty()) {
			cout << "NO\n";
			return 0;
		}
		l = Q.front();
		r = nx[l];
		Q.pop();
		while (!flag[l]) {
			if (Q.empty()) {
				cout << "NO\n";
				return 0;
			}
			l = Q.front();
			r = nx[l];
			Q.pop();
		}
	}
	cout << "YES\n";
	return 0;
}