Say I'm given an oracle that tells me whether or not $L(M)$, the set of words accepted by a Turing Machine $M$, is finite.
By leveraging this oracle, can I solve the halting problem? That is, on an arbitrary input $<M, x>$, can I tell if M is going to halt on $x$ or will it loop forever?
My intuition says no because I don't see a way to leverage the oracle. For example, even with a stronger oracle that tells me if $x \in L(M)$, I still don't know, for inputs $x \notin L(M)$, how $M$ is going to behave. It could either reject (and halt), or loop forever.
Hence, I can't seem to solve the problem even with a stronger oracle that computes $L(M)$ explicitly in case it's finite. So, I believe merely knowing $L(M)$ is finite or not doesn't lead to a solution for the halting problem.
Is my reasoning correct, or am I overlooking something here?