CMO 2019 Problem 1

Amy has drawn three points in a plane, $A, B$, and $C$, such that $A B=B C=C A=6$. Amy is allowed to draw a new point if it is the circumcenter of a triangle whose vertices she has already drawn. For example, she can draw the circumcenter $O$ of triangle $A B C$, and then afterwards she can draw the circumcenter of triangle $A B O$.

(a) Prove that Amy can eventually draw a point whose distance from a previously drawn point is greater than 7 .

(b) Prove that Amy can eventually draw a point whose distance from a previously drawn point is greater than 2019.

(Recall that the circumcenter of a triangle is the center of the circle that passes through its three vertices.)