Today you've found yourself standing on an infinite 2D plane at coordinates (X0, Y0). There are also N targets on this plane, with the ith one at coordinates (Xi, Yi).
You have a boomerang which you can throw in a straight line in any direction from your initial location. After you throw it, you may instantaneously run to any location on the plane. After the boomerang has travelled a distance of exactly D along its initial trajectory, it will return directly to you — that is, to your chosen final location. Note that you cannot move around once the boomerang has started its return trip — its path will always consist of 2 line segments (the first of which has a length of exactly D). The boomerang and the targets have infinitesimal size.
Let A be the number of targets which your boomerang hits (directly passes through) during the first segment of its flight, and B be the number of targets which it hits during the second segment. Your throw is then awarded a score of A * B. What's the maximum score you can achieve? Note that, if there is a target at the exact location at which the two segments meet (at a distance of D from your initial location), then it counts towards both A and B!
Input begins with an integer T, the number of planes. For each plane, there is first a line containing the space-separated integers X0 and Y0. The next line contains the integer D, and the one after contains the integer N. Then, N lines follow, the ith of which contains the space-separated integers Xi and Yi.
For the ith plane, print a line containing "Case #i: " followed by the maximum score you can achieve.
1 ≤ T ≤ 20
1 ≤ N ≤ 3,000
1 ≤ D ≤ 100
-100 ≤ Xi, Yi
≤ 100, for 0 ≤ i ≤ N
All coordinates are pairwise distinct. The following restrictions are also guaranteed to hold for the input given:
For any three targets at distinct points a, b, and c, it is guaranteed that c is either closer than 10-13 away from the infinite line between a and b (and is considered to be on the line), or is further than 10-6 away (and is considered to not be on the line).
Let p be any point at which the boomerang may change direction after hitting a target. For any two targets at distinct points a and b, it is guaranteed that p is either closer than 10-13 away from the infinite line between a and b (and is considered to be on the line), or is further than 10-6 away (and is considered to not be on the line).
On the first plane, one optimal strategy is to throw the boomerang in the direction of the positive x-axis (that is, to (6, 0)), and then run to (0, 0). It will hit targets 2 and 3 on the first segment of its flight, and all 3 targets on the second segment, for a score of 2*3=6.