Pigeonhole and Extremal Arguments · Challenge problem

Markov triples solve $x^2 + y^2 + z^2 = 3xyz$ in positive integers. Replacing the largest entry $z$ by $3xy - z$ gives another solution with a SMALLER largest entry, so descent ends at $(1, 1, 1)$. Starting from $(5, 13, 194)$, how many replacements reach $(1, 1, 1)$?