Platinum and Taxane Chemo in Met Castration Resistant Prostate Cancer Patients With Alterations in DNA Damage Response Genes

Inclusion Criteria: * Histologic diagnosis of adenocarcinoma of the prostate. The presence of neuroendocrine or small cell carcinoma will be exclusionary * Prior treatment with any ARPI, such as abiraterone, enzalutamide, apalutamide, or darotlutamide, is required. * Participants must have recovered to ≤ grade 1 from all reversible toxicity related to prior systemic or radiation therapy, with the exception of chemotherapy induced alopecia and grade 2 peripheral neuropathy, and have adequate washout as follows: * Longest of one of the following: * Two weeks for oral therapies (such as abiraterone, apalutamide, enzalutamide, darolutamide and olaparib) * Standard cycle of standard IV or IM therapies (such as radium 223 or Lu-177-PSMA-617 * 2 weeks, 5 half lives, or standard cycle length (whichever is longer) for investigational agents * Previous major surgery is permitted provided that surgery occurred at least 28 days prior to participant enrollment and that wound healing has occurred * Prior external beam radiation is permitted provided a minimum of 7 days have elapsed between the last dose of radiation and date of enrollment * Radiologically documented presence of metastatic disease within 28 days prior to randomization * Disease progression after ARPI therapy as assessed by the investigator with at least one of the following: * PSA progression with a minimum of two rising PSA values at least 1 week apart, at least 25% and 2ug/L above baseline/nadir. * Radiographic pr…