Fano–Enriques varieties of genera 14, 15, and 16

We will talk about Fano-Enriques threefolds (threefolds having ample anticanonical class and whose general hypersurfaces are smooth Enriques surfaces). It's well known that genus of such threefolds <=17 and there is only one Fano–Enriques threefold with genus 17. I will prove that genus can't be equal to 14, 15, and 16.