In the ever-popular role-playing game Dungeons & Dragons, or D&D, some wizards enhance their skills by training as geometers, specialist spellcasters who rely on the inherent magical properties of geometric forms.
Unlike low-level wizards who may be able to fight only a single zombie at a time, you, the budding geometer, are ready and able to fight medium-sized hordes!
There are N zombies on an infinite 2D plane, with the _i_th one at coordinates (Xi, Yi). You have two spells at your disposal, but you can only cast each one once.
You'll first cast the function-spell Move(x, y, r, dx, dy). Each of its five
parameters can be any real number, though r must be positive. Upon
performing this spell, each zombie initially within the circle centered at
coordinates (x, y) and having radius r will have its coordinates
translated by (dx, dy). In other words, it will have dx added to its
x-coordinate, and dy added to its y-coordinate. Zombies on the border of the
circle are also affected.
You'll then cast the function-spell Destroy(x, y). Each of its two
parameters can be any real number. Upon performing this spell, each zombie
currently within the axis-aligned square centered at coordinates (x, y)
and having side-length R will be destroyed. Zombies on the border of the
square are also affected.
What's the largest number of zombies that can be destroyed by this combination of function-spells?
Input
Input begins with an integer T, the number of zombie hordes you fight. For each case, there is first a line containing two space-separated integers, N and R. Then, N lines follow, the _i_th of which contains 2 space-separated integers, Xi and Yi.
Output
For the _i_th case, print a line containing "Case #i: " followed by the maximum number of zombies you can slay with your spells.
Constraints
1 ≤ T ≤ 50
1 ≤ N ≤ 50
0 ≤ Xi, Yi ≤ 1,000,000,000
1 ≤ R ≤ 1,000,000,000
It is guaranteed that no two zombies are initially at the same point, though zombies may later be moved such that there are multiple at the same coordinates.
Explanation of Sample
In the first case (shown in the pictures above), one optimal strategy is to
first cast Move(2.5, 4.5, 2.2, 3, -1), moving 4 zombies, and then cast
Destroy(5.5, 2.5), destroying 6 zombies.