Color-composite image (credits https://esahubble.org/images/heic1416a/) of the galaxy cluster M0416. White circles show the sample of 182 multiple images in common with the lens model by Bergamini et al. (2021). The 55 new images added to the previous ones and used to optimize the lens model described in this work are plotted in red. A blue circle indicates the galaxy optimized separately from the cluster member scaling relations (see Sect. 3). Colored squares highlight the systems of multiple images analyzed using a new forward modeling approach (described in Sect. 4) that provides a further validation of our lens model. The two BCGs (BCG-N and BCG-S) and a foreground galaxy at z = 0.112 are encircled in gray.