Prostate Specific Membrane Antigen (PSMA) or (FACBC) PET/CT Site-Directed Therapy for Treatment of Prostate Cancer, Flu-BLAST-PC Study

OUTLINE: Patients are assigned to 1 of 3 groups. GROUP I: Patients for whom initial fluciclovine or PSMA PET/CT does not reveal any abnormalities outside the prostatic fossa undergo PSA rechecks every 3 months, and undergo fluciclovine or PSMA PET/CT once PSA is \> 2 ng/ml. If still no abnormalities are found outside of the prostatic fossa, patients continue to undergo PSA rechecks every 3 months, and undergo fluciclovine or PSMA PET/CT once PSA is \> 5 ng/ml. Patients are off study for treatment plan once PSA reaches 10 ng/ml. GROUP II: Patients undergo fluciclovine or PSMA PET/CT and who have =\< 3 regions of metastatic disease outside of the prostatic fossa that are amenable to metastasis-directed therapy undergo lymphadenectomy or radiation therapy. Six to ten weeks after surgery, patients receive abiraterone acetate 1000 mg orally (PO) once daily (QD) and prednisone PO QD. Treatment repeats every 4 weeks for 6 cycles in the absence of disease progression or unacceptable toxicity. Patients may start radiation therapy after 2 cycles of abiraterone acetate and prednisone. GROUP III: Patients undergo fluciclovine or PSMA PET/CT and who have \> 3 regions of metastatic disease receive abiraterone acetate and prednisone as in Group II. After completion of study treatment, patients are followed up at 37, 49, 61, 73, 85, 97, 109, and 121 weeks, at 36, 42, 48, 54, 60, and 66 months, and then annually thereafter.