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The first James Webb Space Telescope (JWST) data on the massive colliding cluster El Gordo allow for 23 known families of multiply lensed images to be confirmed and for eight new members of these families to be identified. Based on these families, which have been confirmed spectroscopically by MUSE, we derived an initial lens model. This model guided the identification of 37 additional families of multiply lensed galaxies, among which 28 are entirely new systems, and nine were previously known. The initial lens model determined geometric redshifts for the 37 new systems. The geometric redshifts agree reasonably well with spectroscopic or photometric redshifts when those are available. The geometric redshifts enable two additional models that include all 60 families of multiply lensed galaxies spanning a redshift range 2 < z < 6. The derived dark-matter distribution confirms the double-peak configuration of mass found by earlier work with the southern and northern clumps having similar masses. We confirm that El Gordo is the most massive known cluster at z > 0.8 and has an estimated virial mass close the maximum mass allowed by standard cosmological models. The JWST images also reveal the presence of small-mass perturbers that produce small lensing distortions. The smallest of these is consistent with being a dwarf galaxy at z = 0.87 and has an estimated mass of 3.8× 109 Mamiddot;, making it the smallest substructure found at z > 0.5. The JWST images also show several candidate caustic-crossing events. One of them is detected at high significance at the expected position of the critical curve and is likely a red supergiant star at z = 2.1878. This would be the first red supergiant found at cosmological distances. The cluster lensing should magnify background objects at z > 6, making more of them visible than in blank fields of a similar size, but there appears to be a deficiency of such objects.