Great Picard's theorem and cardinality

Question

Consider a function $f:\mathbb C\to\mathbb C$ with an essential singularity at $\infty.$ Let $\alpha,\beta$ be two distinct complex "non-exceptional" numbers and consider the sets $$A=\{z\in\mathbb C\mid f(z)=\alpha\},\quad B=\{z\in\mathbb C\mid f(z)=\beta\}.$$ Can it happen that $|A|=\aleph_0,|B|=c$? Or are the level sets always either countable or uncountable?

Answer 1

If the set is uncountable then $f$ assumes $\alpha$ or $\beta$ on an open connected set which eventually turns $f$ into a constant function (Identity Theorem). This contradicts that $f$ has an essential sigularity at $\infty$.

Answer 2

Great Picard's Theorem states that there is at most one $w_0\in\mathbb{C}$ such that for all $w\not =w_0$ and every punctured neighborhood $U$ of $\infty$ the cardinality of $U\cap f^{-1}(w)$ is infinite.
The cardinality of $U\cap f^{-1}(w_0)$ can be finite, think of $f(z)=p(z)\cdot e^z$ for a non-constant polynomial $p$.