Start with the picture. A teenager on a single acne drug for two years now has nails that look blue black. So the question wants the acne antibiotic famous for abnormal pigment deposits.
Acne is treated long term with the tetracycline family, so the answer should sit in that group. Within it, minocycline stands apart. It is lipophilic, reaches many tissues, and forms complexes with melanin and iron. These complexes settle as blue black or grey patches in nails, sclera, gums, teeth, scars, and skin. The history of years of use fits this slow buildup.
Rule out the look-alikes. Doxycycline and tetracycline share the class but are linked mainly to sun sensitivity and to yellow tooth staining in small children, not blue black nails. Azithromycin is a macrolide given in short bursts and has no such pigment effect.
So the single drug that explains acne therapy plus blue black nail pigmentation is minocycline. The pigment usually lightens over months once the drug is stopped.
\[\boxed{\text{Minocycline}}\]