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 Haskell (GHC 7.10.3) (judge verdict: Time Limit Exceeded)

import qualified Data.Map as M
import Data.List

solve :: [[Int]] -> String
solve ((w:h:n:_):b)
  | stack == [0] = "YES"
  | otherwise = "NO"
  where
    edge x y = x == 0 || x == w || y == 0 || y == h
    clockwise a b = compare (toLiner a) (toLiner b)
      where toLiner (i,(x,y))
              | y == 0 = x
              | x == w = w + y
              | y == h = w + h + w - x
              | otherwise = w + h + w + h - y

    sortIndices l = map fst $ sortBy clockwise ps
      where ps = concatMap (\(i, x1:y1:x2:y2:_) -> [(i,(x1,y1)), (i,(x2,y2))]) $ zip [1..] l

    b' = sortIndices $ filter (\(x1:y1:x2:y2:_) -> edge x1 y1 && edge x2 y2) b
    stack = foldl' (\(z:zs) i -> if i /= z then i : z : zs else zs) [0] b'

main = getContents >>= putStrLn.solve.map (map read . words).lines