Inclusion Criteria:  Male 18 years or older able to understand and provide signed written informed consent. Histologically or pathologically confirmed prostate adenocarcinoma without predominant small cell component.  Progressive disease by one or more of the following criteria:  Serum/plasma PSA progression defined as 2 consecutive increases in PSA over a previous reference value measured at least 1 week apart with a minimum start value of >2 ng/mL. Progression of measurable disease (RECIST 1.1) or presence of at least two new bone lesions (PCWG3 criteria).  Previous treatment with next-generation androgen receptor (AR)-directed therapy (e.g. abiraterone, enzalutamide, apalutamide, darolutamide).  Must have received no more than one previous AR-directed therapy. Must have been administered ARAT (abiraterone, enzalutamide, darolutamide, or apalutamide) in the castration-sensitive or castration-resistant setting. Must have progressed while on ARAT. PSMA-PET scan (e.g., 68Ga-PSMA-11 or 18F-DCFPyL) positive as determined by central reader. Effective castration with serum testosterone level of <50 ng/dL and plan to continue with chronic medical or surgical castration. Ability to attend required study visits and return for adequate follow-up, as required by this protocol. Patients with HIV that are healthy and with a low risk of acquired immune deficiency syndrome related outcomes may participate in the trial at the investigators' discretion. Patients with HBV and HCV may also participate if symptoms are sufficiently managed. Life expectancy of at least 6 months as assessed by investigator. Willing to initiate ARAT therapy determined by investigator. For patients who have partners of childbearing potential: The patient and/or partner must use a method of birth control with adequate barrier protection, deemed acceptable by the principal investigator during the study and for 6 months after the last study drug administration.  Exclusion Criteria:  Prior treatment with radioligand therapy including other lutetium-labeled compounds. Prior treatment with radium-223 (Xofigo) within the past 12 weeks. Prior chemotherapy treatment for castration-sensitive or castration-resistant prostate cancer (docetaxel or cabazitaxel). Eastern Cooperative Oncology Group (ECOG) performance status (PS) ≥ 2 Patients with known HRR gene-mutation who have not been previously treated with olaparib or rucaparib. Other concurrent cytotoxic chemotherapy, immunotherapy, radioligand therapy, or investigational therapy.  Inadequate organ and bone marrow function as evidenced by:  Hemoglobin < 8 g/dL. Absolute neutrophil count < 1.5 x 109/L. Platelet count < 100 x 109/L. AST/SGOT and/or ALT/SGPT > 3.0 x ULN. Total bilirubin > 2 x ULN unless patient has known Gilbert's syndrome and then may be 3 x ULN. Creatinine clearance (CrCl) < 50 mL/min based on the Cockcroft-Gault equation. Albumin ≥ 2.75 g/dL Patients who undergo a transfusion for the sole purpose of meeting eligibility for this trial will be excluded. Assessment by the Investigator as unable or unwilling to comply with the requirements of the protocol. Use of an investigational therapeutic drug within the last 4 weeks prior to start of study treatment or scheduled to receive one during the study period. Known CNS metastasis unless received therapy, asymptomatic and neurologically stable. Patients receiving zoledronic acid for bone-targeted therapy must be on stable dose for 4 weeks prior to randomization. Major surgery within 30 days of randomization as determined by the Investigator.  Patients with active significant cardiac disease defined by any of the following:  New York Heart Association class 3 or 4 congestive heart failure within 6 months of signing the ICF unless treated with improvement. Current diagnosis of electrocardiogram abnormalities with significant cardiac arrhythmias History of long QT syndrome or know history of Torsades de Pointe History of myocardial infarction, angina pectoris, or coronary artery bypass graft within 6 months of ICF signature Participants with symptomatic cord compression or clinical/radiological findings indicating impending spinal cord compression Patients with a superscan seen on baseline bone scan as determined by investigator. Active malignancy other than low-grade non-muscle-invasive bladder cancer and non-melanoma skin cancer Previous use of G-CSF for persistent neutropenia after standard of care treatment. Participants who have a pregnant partner or are capable of fathering a child and who are unwilling to take precautions to prevent potential harm to the fetus or prevent pregnancy. Participants with active Covid19. Recovered patients may be included when completely recovered (no symptoms at least 28 days before study medication and a negative Covid test within 72 hours).