  We provide a new construction of rationality for cubic fourfolds via Mori's
theory and the minimal model program. As an application, we present the
solution of the Kuznetsov's conjecture for $d=42$ (the first open case). Our
methods also show an explicit connection between the rationality of cubic
fourfolds belonging to the first four admissible families $\mathcal C_d$, with
$d=14,26,38$, and $42$ and some birational models of minimal K3 surfaces of
degree $d$ contained in well known rational Fano fourfolds.
