Conditional Probability, Independence & Bayes · Superchallenge problem

An integer $X$ is chosen uniformly at random from $\{1,2,\dots,12\}$. Let $B$ be the event that $X$ is a multiple of $3$, and $C$ the event that $X$ is even. An event is any subset $A$ of $\{1,2,\dots,12\}$, including $\varnothing$ and the whole set. For how many events $A$ is $A$ independent of $B$ and also independent of $C$? (Events $E$ and $F$ are independent when $P(E\cap F)=P(E)\,P(F)$.)