  Gravitational lensing by galaxy clusters involves hundreds of galaxies over a
large redshift range and increases the likelihood of rare phenomena
(supernovae, microlensing, dark substructures, etc.). Characterizing the mass
and light distributions of foreground and background objects often requires a
combination of high-resolution data and advanced modeling techniques. We
present the detailed analysis of El Anzuelo, a prominent quintuply imaged dusty
star forming galaxy ($z_{\rm s}=2.29$), mainly lensed by three members of the
massive galaxy cluster ACT-CL$\,$J0102$-$4915, also known as El Gordo ($z_{\rm
d}=0.87$). We leverage JWST/NIRCam data containing previously unseen lensing
features using a Bayesian, multi-wavelength, differentiable and GPU-accelerated
modeling framework that combines Herculens (lens modeling) and NIFTy (field
model and inference) software packages. For one of the deflectors, we
complement lensing constraints with stellar kinematics measured from VLT/MUSE
data. In our lens model, we explicitly include the mass distribution of the
cluster, locally corrected by a constant shear field. We find that the two main
deflectors (L1 and L2) have logarithmic mass density slopes steeper than
isothermal, with $\gamma_{\rm L1} = 2.23\pm0.05$ and $\gamma_{\rm L2} =
2.21\pm0.04$. We argue that such steep density profiles can arise due to
tidally truncated mass distributions, which we probe thanks to the cluster
lensing boost and the strong asymmetry of the lensing configuration. Moreover,
our three-dimensional source model captures most of the surface brightness of
the lensed galaxy, revealing a clump of at most $400$ parsecs at the source
redshift, visible at wavelengths $\lambda_{\rm rest}\gtrsim0.6$ $\mu$m.
Finally, we caution on using point-like features within extended arcs to
constrain galaxy-scale lens models before securing them with extended arc
modeling.
