Effective cyclic membership in a free product

Suppose $P$ has a soluble word problem for a group and $z$ generates an infinite cyclic group. In $P\ast ⟨z⟩$, for $g\ne 1$ the words $gz$ and $[g,z]$ are cyclically reduced with two and four syllables respectively. Their nonzero powers have syllable lengths $2|k|$ and $4|k|$. Given an input's reduced free product form, its length bounds the possible exponents; compare the finitely many candidates using the base word algorithm. Membership and the exponent in each of these infinite cyclic subgroups are therefore computable, with no algorithm for the order of $g$ required.