SHARYGIN 2014 Problem 104

Let $ABC$ be a fixed triangle in the plane. Let $D$ be an arbitrary point in the plane. The circle with center $D$ , passing through $A$ , meets $AB$ and $AC$ again at points $A_b$ and $A_c$ respectively. Points $B_a, B_c, C_a$ and $C_b$ are defined similarly. A point $D$ is called good if the points $A_b, A_c,B_a, B_c, C_a$ , and $C_b$ are concyclic. For a given triangle $ABC$ , how many good points can there be?

(A. Garkavyj, A. Sokolov )