Terminal quartic 3-folds (following A.-S. Kaloghiros's thesis)

We'll obtain rather mild classification of terminal Gorenstein non-factorial quartics in $\mathbb{P}^4$. We'll also prove that they are rational in some cases (for instance, if quartic contains a plane, and its Weil divisors are bounded). Some numerical Sarkisov links appearing in the proof will be supported by geomteric constructions.