Transmitters

In a wireless network where several transmitters share the same frequencies, their signals must not overlap or conflict. One way to achieve this is to limit a transmitter's coverage area. This problem uses a shielded transmitter that broadcasts only over a semicircle.

A transmitter $T$ sits at a fixed location on a $1000 \times 1000$ grid. It broadcasts over a semicircular region of radius $r$ (a half-disk centered at $T$). The transmitter may be rotated by any angle about its position, but it cannot be moved. Given $N$ points on the grid, determine the maximum number of points that the transmitter's signal can cover at the same time. The figure below shows one set of points reached under two different rotations of the transmitter.

The input contains one or more independent transmitter scenarios.

Each scenario starts with a line holding the transmitter's coordinates $x$ and $y$ followed by the broadcast radius $r$. The next line contains the number of points $N$, followed by $N$ lines, each giving the $x$ and $y$ coordinates of one point.

All point coordinates are integers between $0$ and $1000$. The radius $r$ is a positive real number. A point lying exactly on the boundary of the semicircle (its straight edge or its arc) counts as covered. Each scenario has between $1$ and $150$ distinct points, and no point coincides with the transmitter.

The input ends with a line whose radius is negative; on that final line the $x$ and $y$ values are present but meaningless.

For each transmitter scenario, print a single line containing the maximum number of points that can lie within some semicircle.