This phase III trial studies how well standard systemic therapy with or without definitive treatment (prostate removal surgery or radiation therapy) works in treating participants with prostate cancer that has spread to other places in the body. Addition of prostate removal surgery or radiation therapy to standard systemic therapy for prostate cancer may lower the chance of the cancer growing or spreading.

This phase II trial evaluates whether genetic testing in prostate cancer is helpful in deciding which study treatment patients are assigned. Patient cancer tissue samples are obtained from a previous surgery or biopsy procedure and tested for deoxyribonucleic acid (DNA) and ribonucleic acid (RNA) abnormalities or mutations in their cancer. Valemetostat tosylate is in a class of medications called EZH1/EZH2 inhibitors. It blocks proteins called EZH1 and EZH2, which may help slow or stop the spread of tumor cells. Carboplatin is in a class of medications known as platinum-containing compounds. It works in a way similar to the anticancer drug cisplatin, but may be better tolerated than cisplatin. Carboplatin works by killing, stopping or slowing the growth of tumor cells. Cabazitaxel injection is in a class of medications called microtubule inhibitors. It works by slowing or stopping the growth of tumor cells. Abiraterone acetate blocks tissues from making androgens (male hormones), such as testosterone. This may cause the death of tumor cells that need androgens to grow. It is a type of anti-androgen. Enzalutamide is in a class of medications called androgen receptor inhibitors. It works by blocking the effects of androgen (a male reproductive hormone) to stop the growth and spread of tumor cells. Lutetium Lu 177 vipivotide tetraxetan is in a class of medications called radiopharmaceuticals. It works by targeting and delivering radiation directly to tumor cells which damages and kills these cells. Assigning patients to targeted treatment based on genetic testing may help shrink or slow the cancer from growing