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We present a novel application of the extended Fast Action Minimization method (eFAM) aimed at assessing the role of the environment in shaping galaxy evolution. We validate our approach by testing eFAM predictions against the Magneticum hydrodynamical simulation. We consider the z~0 snapshot of the simulation as our observed catalogue and use the reconstructed trajectories of galaxies to model the evolution of cosmic structures. At the statistical level, the fraction of volume (VFF) occupied by voids, sheets, filaments, and clusters in the reconstructed catalogues agrees within $1σ$ with the VFF estimated from the high-redshift snapshots of the simulation. The local accuracy of eFAM structures is evaluated by computing their purity with respect to the simulated catalogues, P, at the cells of a regular grid. Up to z=1.2, clusters have 0.58<P<0.93, filaments vary in 0.90<P<0.99, sheets show 0.78<P<0.92, and voids are best identified with 0.90<P<0.92. As redshift increases, comparing reconstructed tracers and simulated galaxies becomes more difficult due to their different biases and number densities and the purity decreases to P~0.6. We retrieve the environmental history of individual galaxies by tracing their trajectories through the cosmic web and relate their observed gas fraction, $f_\mathrm{gas}$, with the time spent within different structures. For galaxies in clusters and filaments, eFAM reproduces the variation of $f_\mathrm{gas}$ as a function of the redshift of accretion/infall as traced by the simulations with a 1.5 $σ$ statistical agreement (which decreases to 2.5 $σ$ statistical agreement for low-mass galaxies in filaments). These results support the application of eFAM to observational data to study the environmental dependence of observed galaxy properties, offering a complementary approach to that based on light-cone observations.