A < HPL < Complement(A) What can you say about A and Complement(A)?
Your analysis appears to be correct. But some clarity may help you.
Let's analyse each case and check whether it is consistent with given information.
As $\text{HPL} \leq \overline{A}$, $\overline{A}$ cannot be finite or recursive.

\begin{array}{|l|l|}
\hline
\textbf{If A is }  &amp; \textbf{Then }\overline{A} \textbf{ is}\\
\hline
\mbox{Finite} &amp; \mbox{Contradiction - }\overline{A} \text{ has to be FA; Violates HPL} \leq \overline{A} \\
\hline
\mbox{Recursive} &amp; \mbox{Contradiction - }\overline{A} \text{ has to be recursive; Violates HPL} \leq \overline{A}\\
\hline
\mbox{Recursively Enumerable} &amp; \text{Non-Recursively Enumerable}\\
\mbox{but not Recursive} &amp; \\
\hline
\mbox{Non-Recursively Enumerable} &amp; \mbox{Contradiction - Violates } A \leq \text{HPL}\\
\hline
\end{array}

You can now infer countability based on this information.