A Study Evaluating [177Lu]Lu-PSMA-617 vs. a Change of Androgen Receptor-directed Therapy in Taxane Treatment Naive Chinese Male Patients With Progressive Metastatic Castrate Resistant Prostate Cancer

Key Inclusion criteria

1. Participants must be Chinese adult men \>= 18 years of age
2. Participants must have an ECOG performance status of 0 to 1
3. Participant must have histological pathological and/or cytological confirmation of adenocarcinoma of the prostate
4. Participants must be \[68Ga\]Ga-PSMA-11 PET/CT scan positive, and eligible as determined by the sponsor's central reader
5. Participants must have a castrate level of serum/plasma testosterone (\< 50 ng/dl, or \< 1.7 nmol/L)
6. Participants must have progressed only once on prior second generation ARDT (abiraterone, enzalutamide, darolutamide, or apalutamide)) in either HSPC or CRPC setting.

   * first generation androgen receptor inhibitor therapy (e.g. bicalutamide) is allowed but not considered as prior ARDT therapy
   * second generation ARDT must be the most recent therapy received
7. candidates for change in ARDT (eligible to receive abiraterone or enzalutamide) as assessed by the treating physician

   • Participants cannot have previously progressed nor had intolerable toxicity to both enzalutamide and abiraterone
8. Documented progressive mCRPC, based on at least 1 of the following criteria:

   * Serum/plasma PSA progression defined as 2 consecutive increases in PSA measured at least 1 week apart. the minimal start value is 2.0 ng/ml;
   * Soft-tissue progression defined based on PCWG3-modified RECIST v1.1(Eisenhauer et al 2009, Scher et al 2016)
   * Progression of bone disease: two new lesions; only positivity on the bone scan defines metastatic disease to bone (PCWG3 criteria Scher et al 2016)
9. Participants must have at least one metastatic lesion that is present on screening/baseline CT, MRI, or bone scan imaging obtained =\< 28 days prior to randomization
10. Participants must have adequate organ function:

    * Bone marrow reserve:
    * ANC \>= 1.5 x 109/L
    * Platelets \>= 100 x 109/L
    * Hemoglobin \>= 9 g/dL
    * Hepatic:
    * Total bilirubin \< 2 x the institutional upper limit of normal (ULN). For participants with known Gilbert's Syndrome =\< 3 x ULN is permitted
    * ALT or AST =\< 3.0 x ULN OR =\< 5.0 x ULN for participants with liver metastases
    * Albumin \>= 2.5 g/dL
    * Renal:
    * eGFR \>= 50 mL/min/1.73m2 using the Modification of Diet in Renal Disease (MDRD) equation

Key Exclusion criteria

1. Previous treatment with any of the following within 6 months of randomization: Strontium-89, Samarium-153, Rhenium-186, Rhenium-188, Radium-223, Lutitium-177, Actium-225, hemi-body irradiation
2. Previous PSMA-targeted radioligand therapy
3. Prior treatment with PARP inhibitor, cytotoxic chemotherapy for castration resistant or castration sensitive prostate cancer (i.e., taxanes, platinum, estramustine, vincristine, methotrexate, etc.), immunotherapy or biological therapy (including monoclonal antibodies). \[Note: a maximum of 6 cycles of taxane exposure in the adjuvant or neo-adjuvant setting is allowed if 12 months have elapsed since completion of this adjuvant or neo-adjuvant therapy prior to randomization\]
4. Concurrent cytotoxic chemotherapy, immunotherapy, radioligand therapy, PARP inhibitor, biological, or investigational therapy
5. Transfusion or use of bone marrow stimulating agents for the sole purpose of making a participant eligible for study inclusion
6. Participants with a history of CNS metastases who are neurologically unstable, symptomatic, or receiving corticosteroids for the purpose of maintaining neurologic integrity.

   * Participants with CNS metastases are eligible if received therapy (surgery, radiotherapy, gamma knife), asymptomatic and neurologically stable without corticosteroids.
   * Participants with epidural disease, canal disease and prior cord involvement are eligible if those areas have been treated, are stable, and not neurologically impaired.
7. Symptomatic cord compression, or clinical or radiologic findings indicative of impending cord compression
8. Cardiac or cardiac repolarization abnormality, including any of the following:

   * History of myocardial infarction (MI), angina pectoris, or coronary artery bypass graft (CABG) within 6 months prior to starting study treatment
   * Clinically significant cardiac arrhythmias (e.g., ventricular tachycardia), complete left bundle branch block, high-grade AV block (e.g., bifascicular block, Mobitz type II and third degree AV block) and QTc\>=500.
9. Concurrent serious (as determined by the Principal Investigator) medical conditions, including, but not limited to New York Heart Association class III or IV congestive heart failure, history of congenital prolonged QT syndrome, uncontrolled infection, known active hepatitis B or C or other significant co-morbid conditions that in the opinion of the investigator would impair study participation or cooperation
10. Unmanageable concurrent bladder outflow obstruction or urinary incontinence. Note: Participants with bladder outflow obstruction or urinary incontinence, which is manageable and controlled with best available standard of care (incl. pads, drainage) are allowed.