KöMaL 2018 A714

Consider $n\ge 2$ pairwise disjoint disks $D_1,D_2,\dots,D_n$ on the Euclidean plane. For each $k=1,2,\dots,n$, denote by $f_k$ the inversion with respect to the boundary circle of $D_k$. (Here, $f_k$ is defined at every point of the plane, except for the center of $D_k$.) How many fixed points can the transformation $f_n\circ f_{n-1}\circ\cdots\circ f_1$ have, if it is defined on the largest possible subset of the plane?