Motivated by the quest for experimental procedures capable of controlled manipulation of single atoms on surfaces, we set up a computational strategy that explores the cyclical vertical manipulation of a broad set of single atoms on the GaAs(110) surface. First-principles simulations of atomic force microscope tip-sample interactions were performed considering families of GaAs and Au-terminated tip apexes with varying crystalline termination. We identified a subset of tips capable of both picking up and depositing an adatom (Ga, As, Al, and Au) any number of times via a modify-restore cycle that "resets"the apex of the scanning probe to its original structure at the end of each cycle. Manipulation becomes successful within a certain window of lateral and vertical tip distances that are observed to be different for extracting and depositing each atom. A practical experimental protocol of special utility for potential cyclical manipulation of single atoms on a nonmetallic surface is proposed.