AUSTRIAN-POLISH 1989 Problem 5

Let $A$ be a vertex of a cube $\omega$ circumscribed about a sphere $k$ of radius $1$ . We consider lines $g$ through $A$ containing at least one point of $k$ . Let $P$ be the intersection point of $g$ and $k$ closer to $A$ , and $Q$ be the second intersection point of $g$ and $\omega$ . Determine the maximum value of $AP\cdot AQ$ and characterize the lines $g$ yielding the maximum.