Can we find a line L and a closed convex set S such that $L \cap S = \varnothing$, but each plane containing L intersects S

Question

Question: In $\mathbb{R}^{3}$, Can we find a line L and a closed convex set S with $S \cap L = \varnothing$ such that for each plane $\Pi$, $L \subseteq \Pi$ we have $\Pi \cap S \neq \emptyset$?

How can we find these line $L$ and the closed convex set $S$?