Suppose we have $13$ points in the $xy$-plane such that for any $5$-tuple of points, at least $4$ lie on a circle.
a) Find the greatest integer $M \in \{1, 2, \cdots, 13\}$ such that from these $13$ points, at least $M$ lie on a circle but not necessarily $M+1$ lie on a circle.
b) Prove that your $M$ is optimal.
First, I noticed given any $4$ points that lie on a circle, every other $5$-tuple of points that contain said $4$ points already satisfy the conditions, so for optimal $M$ I assumed that no other point lies on the circle formed by the $4$ points. However, I am unsure of how to proceed from here.