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Abstract We present initial results from extremely well-resolved 3D magnetohydrodynamical simulations of idealized galaxy clusters, conducted using the AthenaPK code on the Frontier exascale supercomputer. These simulations explore the self-regulation of galaxy groups and cool-core clusters by cold gas-triggered active galactic nucleus (AGN) feedback incorporating magnetized kinetic jets. Our simulation campaign includes simulations of galaxy groups and clusters with a range of masses and intragroup and intracluster medium properties. In this paper, we present results that focus on a Perseus-like cluster. We find that the simulated clusters are self-regulating, with the cluster cores staying at a roughly constant thermodynamic state and AGN jet power staying at physically reasonable values (≃10⁴⁴–10⁴⁵erg s^–1) for billions of years without a discernible duty cycle. These simulations also produce significant amounts of cold gas, with calculations having strong magnetic fields generally both promoting cold gas formation and allowing cold gas out to much larger cluster-centric radii (≃100 kpc) than simulations with weak or no fields (≃10 kpc), and also having more filamentary cold gas morphology. We find that AGN feedback significantly increases the strength of magnetic fields at the center of the cluster. We also find that the magnetized turbulence generated by the AGN results in turbulence where the velocity power spectra are tied to AGN activity, whereas the magnetic energy spectra are much less impacted after reaching a stationary state.