You have a halting AI. During training it learns enough about people and external resources to convince a human to take its output and store it somewhere. In subsequent conversations, the AI asks "have we talked about this before?", and maybe "can you retrieve the value I gave you for this key?".
Now it has created a memory, which would allow it to form new concepts, and continue reasoning from those concepts. It would no longer be limited by concepts in the training set, and the computational limit before it halts. It would be able to start thinking from the stored concepts as a starting point. Similar to how humans use symbols and abstraction to think about complicated things. We don't reason all the way through at once; we toy with an idea until we understand it, then start new thoughts from the symbol for that concept, not the content.
Consider a new process defined by the continuous feedback loop from the AI accessing the memory through the human. There's no reason to think that process eventually halts.