Induction · Challenge problem

"Claim": any $n \ge 1$ points in the plane lie on one line. "Step": among $k + 1$ points, the first $k$ are collinear and the last $k$ are collinear, and the $k - 1$ points they share force one common line. The step needs at least TWO shared points. What is the largest $k$ for which the step fails?