Bonxie attack

0.65 When walking on the remote isle of Foula, professor Oak was dive-bombed by many bonxies that were protecting their territories. Those (fortunately, failed) attacks inspired this problem.

0.35

Assume an infinite flat world. There, we have $n$ bonxies, each protecting a disc of radius $r_i$ centered at $(x_i, y_i)$ . Please compute a point protected by the maximum number of bonxies.

Input

Input is all integer numbers, and consists of several cases, each one with $n$ followed by the $n$ triples $x_i$ $y_i$ $r_i$ . You can assume $1 \le n \le 1000$ , that all coordinates are at most $10^9$ in absolute value, that all radii are between 1 and $10^9$ , that each pair of circles of the discs either do not intersect, or intersect at exactly two points, and that there is no point in the plane with more than two circles on it. The input cases have no precision issues.

Output

For every case, print the maximum number of bonxies that can protect a point.

Hint

The expected solution has cost $\Theta(n^2 \log n)$ .

Problem information

Author: Ivan Geffner

Generation: 2026-01-25T11:23:49.789Z