Target Practice

Balloons sit on top of poles stuck into the ground at various points of a large field, and the poles are not all the same height. A shooter moves around the field and fires at the balloons, trying to burst them all with as few shots as possible.

A bullet travels in a perfectly straight line and passes right through a balloon, so a single shot bursts every balloon that lies on that line. By aiming carefully, the shooter can burst several balloons with one shot. The shooter may fire from anywhere and at any height, so a shot can be any straight line in space. Treat each balloon as a single point (bullets may pass through the poles).

Given the balloons in a field, determine the minimum number of shots needed to burst all of them.

The input contains several test cases. Each test case begins with an integer $n$ ($n \le 50$), the number of balloons; $n = 0$ marks the end of the input.

Then follow $n$ triples of integers $x\ y\ h$, giving a balloon at field position $(x, y)$ at height $h$. Every integer is greater than $0$ and at most $100$, and no two balloons share the same position $(x, y)$.

For each test case, print one line:

Target set k can be cleared using only s shots.

where $k$ is the test case number (starting at 1) and $s$ is the minimum number of shots needed to burst all balloons in that set.