(proof-new) Update the strings inference manager for proofs (#5220)

This updates the inference manager in strings in two ways:
(1) It now inherits from InferenceManagerBuffered, meaning that the custom methods for process the current pending lemma/fact vectors are removed in favor of the standard ones.
(2) It has support for proof generation via the InferProofCons utility.

This PR standardizes three methods processFact, processLemma, and processConflict which take InferInfo and processes any string-specific behavior pertaining to processing facts, lemmas and conflicts with the inference manager. These methods are intended to preserve the previous behavior of this class.

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