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++ (GCC 9.2.1) (judge verdict: Accepted)

#pragma GCC optimize("O2,unroll-loops")
#pragma GCC target("avx,avx2,sse,sse2,ssse3")
#include<bits/stdc++.h>
#define int long long
#define all(x) x.begin(), x.end()
using namespace std;
using ll = long long;
using ld = long double;
const int maxn = 1<<20, mod = 1e9 + 7;
mt19937_64 rng(chrono::steady_clock::now().time_since_epoch().count());
int r, c, n;
vector<array<int, 2>> lx, rx, ly, ry;
bool border(int x, int y) {
	return x == 0 || x == r || y == 0 || y == c;
}
void add(int x, int y, int i) {
	if(x == 0) {
		lx.push_back({y, i});
	} else if(x == r) {
		rx.push_back({y, i});
	} else if(y == 0) {
		ly.push_back({x, i});
	} else if(y == c) {
		ry.push_back({x, i});
	}
}
array<ll, 2> h[maxn];
signed main() {
	cin.tie(0)->sync_with_stdio(0);
	cin >> r >> c >> n;
	for(int x, y, u, v, c = 0, i = 0; i < n; i++) {
		cin >> x >> y >> u >> v;
		if(!border(x, y) || !border(u, v)) continue;
		add(x, y, i);
		add(u, v, i);
	}
	vector<int> r;
	int rev = 0;
	for(auto V : {lx, ry, rx, ly}) {
		if(rev > 1)
			sort(all(V), greater<array<int, 2>>());
		else
			sort(all(V), less<array<int, 2>>());
		rev++;
		for(auto &[v, idx] : V) r.push_back(idx);
	}
	for(int i = 0; i < n; i++) h[i][0] = rng(), h[i][1] = rng();
	array<ll, 2> cur = {0, 0};
	vector<array<ll, 2>> ps;
	vector<int> lst(n, -1);
	for(int i = 0; i < r.size(); i++) {
		if(lst[r[i]] != -1) {
			if((ps.back()[0]^ps[lst[r[i]]][0])||(ps.back()[1]^ps[lst[r[i]]][1])) return cout << "NO\n", 0;
		} else lst[r[i]] = i;
		cur[0] ^= h[r[i]][0];
		cur[1] ^= h[r[i]][1];
		ps.push_back(cur);
	}
	cout << "YES\n";
}