Problem D: Where are the Squares?

The Hough Transform (HT) is a method for detecting linear structures in images. It can be used to isolate features of a particular shape within an image. Commonly it is used in image processing to detect lines. Lines can be represented in Cartesian space using the equation

y = mx + c

For computational purposes it is sometimes better to use Polar Coordinates in the equation

y =

⎛
⎝

−

cos θ

sin θ

⎞
⎠

⎛
⎝

sin θ

⎞
⎠

This can be rearranged in the form of a Hough Space (r, θ):

r = x cos θ+ y sin θ

Using this equation every line can be described as a combination of (r, θ), where r is the length of the normal vector that connects the line to the origin and θ is the angle of vector r. In order to be able to represent all possible lines the angle varies in the interval [0^°,180^°[ (values in degrees, not radians) and r is negative whenever it is bellow the abscissa. Example:

Line 1: r₁ = 10, θ₁ = 30^°

Line 2: r₂ = −8, θ₂ = 135^°

Line 3: r₃ = 25, θ₃ = 0^°

Line 4: r₄ = 23, θ₄ = 90^°

Given an image with a set of detected lines in Hough Space (no repeated lines), it is possible to detect all squares in the image as seen in the example:

Task

Given an image with width W, height H and a set of L detected infinite straight lines, return the number of regular squares S in which the center of the square is inside the image. This means that for each square center (cx,cy) the following conditions apply: 0 ≤ cx < W and 0 ≤ cy < H.

Input

The first line consists of three integers, W and H the width and height of the image and L, the number of lines. Then L lines follow, in each line there are two integers r_i and θ_i, representing the Hough parameters of each line.

Output

The output consist of a single line containing S, the number of squares with center inside the image.

Constraints

0 ≤ W,H ≤ 2000
0 ≤ θ_i < 180
0 < L ≤ 1000

Input example 1