Never trust Ivan (2)

Given $n$ points on the plane $p_1 \dots p_n$ no three of which are collinear, find a permutation of the points $p_{i_1} \dots p_{i_n}$ such that the $n$ segments $(p_{i_1}, p_{i_2}), (p_{i_2}, p_{i_3}), \dots, (p_{i_{n-1}}, p_{i_n}), (p_{i_n}, p_{i_1})$ form a non-degenerate polygon, that is, one that does not cross itself.

Input

Input consists of several cases, each one with $n$ followed by $n$ points with integer coordinates not larger than $10^7$ in absolute value. Assume $3 \le n \le 10^4$ , and that no three given points are collinear.

Output

For every case, print a correct polygon constructed from the $n$ points. If there is more than one solution, print any of them. If there is no solution, print “Ivan is a troll”.

Problem information

Author: Ivan Geffner

Generation: 2026-01-25T11:20:49.653Z