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We assemble a catalogue of candidate Sagittarius stream members with 5d and 6d phase-space information, using astrometric data from Gaia DR2, distances estimated from RR Lyrae stars, and line-of-sight velocities from various spectroscopic surveys. We find a clear misalignment between the stream track and the direction of the reflex-corrected proper motions in the leading arm of the stream, which we interpret as a signature of a time-dependent perturbation of the gravitational potential. A likely cause of this perturbation is the recent passage of the most massive Milky Way satellite - the Large Magellanic Cloud (LMC). We develop novel methods for simulating the Sagittarius stream in the presence of the LMC, using specially tailored N-body simulations and a flexible parametrization of the Milky Way halo density profile. We find that while models without the LMC can fit most stream features rather well, they fail to reproduce the misalignment and overestimate the distance to the leading arm apocentre. On the other hand, models with an LMC mass in the range (1.3+-0.3)x10^11 Msun rectify these deficiencies. We demonstrate that the stream can not be modelled adequately in a static Milky Way. Instead, our Galaxy is required to lurch toward the massive in-falling Cloud, giving the Sgr stream its peculiar shape and kinematics. By exploring the parameter space of Milky Way potentials, we determine the enclosed mass within 100 kpc to be (5.6+-0.4)x10^11 Msun, and the virial mass to be (9.0+-1.3)x10^11 Msun, and find tentative evidence for a radially-varying shape and orientation of the Galactic halo.