Circle Artwork

The circle is an ancient and universal symbol of unity, wholeness, and infinity. Playing the role of a modern artist, we want to compose a painting from colored points and circles.

First we place several colored points on the canvas. For each color $C_i$ we would like to draw one circle that satisfies both of the following conditions:

• every colored point lying inside the circle or on its boundary has color $C_i$;
• at least two colored points lie on the boundary of the circle.

Since a point on the boundary is a point "inside or on the boundary", any boundary point must itself have color $C_i$. Hence a valid circle for color $C_i$ passes through at least two points of color $C_i$ and contains no point of any other color, neither strictly inside nor on the boundary. Points of color $C_i$ may lie inside, on, or outside the circle. For some colors no such circle exists.

Given the colored points, determine the largest number of colors for which such a circle exists — that is, how many colors admit at least one valid circle.

The input contains several test cases. Each test case begins with a line containing a single integer $n$ ($1 \le n \le 100$), the number of colored points. Each of the next $n$ lines has the form C X Y, where C is the color of the point (a string of at most $20$ lowercase English letters) and $X$, $Y$ are its integer coordinates with $-1{,}000{,}000 \le X, Y \le 1{,}000{,}000$.

The input ends with a line containing a single $0$.

For each test case, print a single line containing the largest number of colors for which a valid circle exists.