Note that since there are 8 faces to a regular hexagonal prism and a cross-section may only intersect a face once, the upper bound for our answer is 8. Indeed, we can construct a cross-section of the prism with 8 sides. Let $ABCDEF$ and $A'B'C'D'E'F'$ be the two bases of the prism, with $A$ being directly over $A'$. Choose points $P$ and $Q$ on line segments $AB$ and $BC$, respectively, and choose points $P'$ and $Q'$ on segments $D'E'$ and $E'F'$, respectively, such that $PQ \parallel P'Q'$. Then, the cross-section of the prism from the plane that goes through $P, Q, P'$, and $Q'$ forms a polygon with 8 sides.
Compute the maximum number of sides of a polygon that is the cross-section of a regular hexagonal prism.