Standard Systemic Therapy With or Without Definitive Treatment in Treating Participants With Metastatic Prostate Cancer

Inclusion Criteria: * STEP 1 REGISTRATION: DISEASE-RELATED CRITERIA: All patients must have a histologically or cytologically proven diagnosis of adenocarcinoma of the prostate. Patients with pure small cell carcinoma\* (SCC), sarcomatoid, or squamous cell carcinoma are not eligible. (\*morphology must be consistent with SCC; synaptophysin or chromogranin positive by immunohistochemical staining is insufficient to diagnose SCC). * STEP 1 REGISTRATION: DISEASE-RELATED CRITERIA: Patients must have an intact prostate. No prior local therapy for prostate adenocarcinoma is allowed (e.g., brachytherapy, high-intensity focused ultrasound \[HIFU\], cryotherapy, laser ablative therapies). Any prior therapy for benign conditions, such as obstruction, are acceptable (e.g., transurethral resection of the prostate, greenlight laser ablation, microwave ablation). * STEP 1 REGISTRATION: DISEASE-RELATED CRITERIA: Patients must have evidence of metastatic disease on technetium bone scan and computed tomography (CT) or magnetic resonance imaging (MRI) within 42 days prior to starting standard systemic therapy. Metastatic disease that is detected by positron emission tomography (PET) scan only (sodium fluoride \[NaF\], prostate-specific membrane antigen \[PSMA\], anti-1-amino-3-18F-fluorocyclobutane-1-carboxylic acid \[FACBC\], carbon \[C\]11) but not conventional imaging (technetium \[Tc\]99 bone scan, CT or MRI) or solitary metastases by conventional imaging, must be confirmed histologically…