KöMaL 2021 B5184

Cornelia marked four non-concyclic points in the plane. Then she drew all the circles that pass equidistant from the four points. What is the maximum possible number of circles she may have drawn? (The distance between point $P$ and a circle $k$ centred at $O$ is defined as follows: let $M$ denote the point where the ray starting at $O$ and passing through $P$ intersects the circle $k$. Then the distance is the length of line segment $PM$.)