SMO 2018 Problem 1

Consider a regular cube with side length $2$ . Let $A$ and $B$ be $2$ vertices that are furthest apart. Construct a sequence of points on the surface of the cube $A_1$ , $A_2$ , $\ldots$ , $A_k$ so that $A_1=A$ , $A_k=B$ and for any $i = 1,\ldots, k-1$ , the distance from $A_i$ to $A_{i+1}$ is $3$ . Find the minimum value of $k$ .