Why can't causal loops exist? (Second Way)

Aquinas states in his Second Way that nothing can be the efficient cause of itself, for then it would have to exists prior to itself, and that's just absurd. I accept this. If something was the efficient cause of itself (self-caused), then there must have been a time when that thing was both existing and not-existing at the same time. This is a clear violation of the Law of Non-contradiction.

However, I don't understand why something can't be causally responsible for its own existence, in a cyclical fashion. Consider the following:

A --> B --> C --> A

Here, A is the efficient or immediate cause for B, but it is also causally responsible for the existence of C and itself. Every member in this cycle is causally responsible for the existence of every other member including itself. Therefore, there doesn't need to be any first, uncaused cause.

So why isn't this a possibility, why must their exist a uncaused first cause?