Photo Shoot

Adam Ansels is a photographer who specializes in impromptu photos of his clients. Right now Adam is standing in the middle of a field, surrounded by a large group of people.

Adam's camera has a fixed field-of-view angle $f$: if he points the camera in a direction $d$ (measured in degrees from the $x$-axis), then everything in the range from $d - f/2$ to $d + f/2$ appears in the picture.

Adam wants to take as few pictures as possible. Given the locations of the people around Adam and the camera's field-of-view angle, determine the minimum number of photos Adam must take so that everyone appears in at least one photo.

Each test case starts with a line containing four integers $n$, $x$, $y$, $f$: the number of people surrounding Adam ($n \ge 0$), Adam's location $(x, y)$, and the field-of-view of his camera in degrees ($f > 0$). The maximum value of $n$, $|x|$, and $|y|$ is $100$, and the maximum value of $f$ is $180$.

This is followed by $n$ coordinate pairs $x_i\ y_i$ giving the locations of the $n$ people ($|x_i|, |y_i| \le 1000$). No two people (including Adam) stand in the same spot. All locations use the standard Cartesian $x$-$y$ coordinate system.

A line consisting of four zeros terminates the input.

For each test case, output the case number followed by the minimum number of photos Adam needs so that everyone appears in at least one picture. You may assume that no two people are exactly $f$ degrees apart from each other relative to Adam. Print each answer in the form Case k: x, where $k$ is the case number starting from 1 and $x$ is the minimum number of photos.