Proof Writing & Problem Solving · Superchallenge problem

A ring puzzle has $10$ rings in a row, numbered $1$ to $10$, each either ON or OFF. Ring $1$ can always be switched (ON to OFF or back). For $k \ge 2$, ring $k$ can be switched only when ring $k - 1$ is ON and rings $1, \ldots, k - 2$ are all OFF. One move switches one ring. Starting with all rings ON, what is the smallest number of moves that turns all of them OFF?