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The Magellanic Stream and the Leading Arm form a massive, filamentary system of gas clouds surrounding the Large and Small Magellanic Clouds. Here we present a new component-level analysis of their ultraviolet (UV) kinematic properties using a sample of 31 sightlines through the Magellanic System observed with the Hubble Space Telescope/Cosmic Origins Spectrograph. Using Voigt profile fits to UV metal-line absorption, we quantify the kinematic differences between the low-ion (Si II and C II), intermediate-ion (Si III), and high-ion (Si IV and C IV) absorption lines and compare the kinematics between the Stream and Leading Arm. We find that the Stream shows generally simple, single-phase kinematics, with statistically indistinguishable b-value distributions for the low-, intermediate-, and high-ion components, all dominated by narrow (b<25 km/s) components that are well aligned in velocity. In contrast, we find tentative evidence that the Leading Arm shows complex, multi-phase kinematics, with broader high ions than low ions. These results suggest that the Stream is photoionized up to C IV by a hard ionizing radiation field. This can be naturally explained by the Seyfert-flare model of Bland-Hawthorn et al. (2013, 2019), in which a burst of ionizing radiation from the Galactic Center photoionized the Stream as it passed below the south Galactic pole. The Seyfert flare is the only known source of radiation that is both powerful enough to explain the H-alpha intensity of the Stream and hard enough to photoionize Si IV and C IV to the observed levels. The flare's timescale of a few Myr suggests it is the same event that created the giant X-ray/gamma-ray Fermi Bubbles at the Galactic Center.