  We present a detailed strong lensing analysis of the massive and distant
($z=0.870$) galaxy cluster ACT-CL J0102$-$4915 (ACT0102, also known as El
Gordo), taking advantage of new spectroscopic data from the Multi Unit
Spectroscopic Explorer (MUSE) on the Very Large Telescope, and archival imaging
from the Hubble Space Telescope. Thanks to the MUSE data, we measure secure
redshifts for 374 single objects, including 23 multiply lensed galaxies, and
167 cluster members of ACT0102. The observed positions of 56 multiple images,
along with their new spectroscopic redshift measurements, are used as
constraints for our strong lensing model. Remarkably, some multiple images are
detected out to a large projected distance of $\approx 1$ Mpc from the
brightest cluster galaxy, allowing us to estimate a projected total mass value
of $1.84_{-0.04}^{+0.03} \times 10^{15}\, \rm M_{\odot}$ within that radius. We
find that we need two extended cluster mass components, the mass contributions
from the cluster members and the additional lensing effect of a foreground
($z=0.633$) group of galaxies, to predict the positions of all multiple images
with a root mean square offset of $0.75"$. The main cluster-scale mass
component is centered very closely to the brightest cluster galaxy and the
other extended mass component is located in the north-west region of the
cluster. These two mass components have very similar values of mass projected
within 300 kpc from their centers, namely
$2.29_{-0.10}^{+0.09}\times10^{14}\,\rm M_{\odot}$ and
$2.10_{-0.09}^{+0.08}\times10^{14}\,\rm M_{\odot}$, in agreement with the major
merging scenario of ACT0102. We make publicly available the lens model,
including the magnification maps and posterior distributions of the model
parameter values, as well as the full spectroscopic catalogue containing all
redshift measurements obtained with MUSE.
