Cabazitaxel +/- Carboplatin vs 177Lu-PSMA-617 in Metastatic Castrate-resistant Prostate Cancer

The purpose of this study is to find out what treatment works best for participants with metastatic prostate cancer that are not responding to hormone treatment and docetaxel and are also Prostate-specific membrane antigen(PSMA) positive.

Inclusion Criteria: * Participants must have histologically or cytologically confirmed adenocarcinoma of prostate * Evidence of metastatic castrate-resistant prostate cancer that has previously been treated with an androgen receptor pathway inhibitor. Prior docetaxel exposure is recommended but not mandatory. Tissue is not mandatory, but a pathologic report is required at time of enrollment. * Patients must have a PSMA-positive 18F-rhPSMA-7.3 performed within 12 weeks from C1D1 with ≥1 site with SUVmax ≥10. An alternative PSMA PET tracer is permitted at baseline if performed within 8 weeks prior to randomization. * Eligible patients have evidence of mCRPC who have progressed on prior novel hormonal agent(s) to include at least one of the following: * Baseline PSMA SUVmean \ 18 years. * ECOG performance status of 0 to 2. * Participants must have adequate organ and marrow function as defined below to be suitable for the randomized treatment outlined in this * Absolute neutrophil count \>1000/μL; platelet count \>90 000/μL; hemoglobin \>8.5 g/dL) at screening. * Note: Participants must not have received any growth factors within 7 days or blood transfusions within 14 days prior to the hematologic laboratory values obtained at screening). * Total bilirubin (TBIL) \