  We present N-body simulations of a Sagittarius-like dwarf spheroidal galaxy
(Sgr) that follow its orbit about the Milky Way (MW) since its first crossing
of the Galaxy's virial radius to the present day. As Sgr orbits around the MW,
it excites vertical oscillations, corrugating and flaring the Galactic stellar
disc. These responses can be understood by a two-phase picture in which the
interaction is first dominated by torques from the wake excited by Sgr in the
MW dark halo before transitioning to tides from Sgr's direct impact on the disc
at late times. We show for the first time that a massive Sgr model
simultaneously reproduces the locations and motions of arc-like over densities,
such as the Monoceros Ring and the Triangulum Andromeda stellar clouds, that
have been observed at the extremities of the disc, while also satisfying the
solar neighbourhood constraints on the vertical structure and streaming motions
of the disc. In additional simulations, we include the Large Magellanic Cloud
(LMC) self consistently with Sgr. The LMC introduces coupling through
constructive and destructive interference, but no new corrugations. In our
models, the excitation of the current structure of the outer disk can be traced
to interactions as far back as 6-7 Gyr ago (corresponding to $z\leq1$). Given
the apparently quiescent accretion history of the MW over this timescale, this
places Sgr as the main culprit behind the vertical oscillations of the disc and
the last major accretion event for the Galaxy with the capacity to modulate its
chemodynamical structure.
