This study is for patients who have been diagnosed with prostate cancer that is prostate-specific membrane antigen (PSMA)-positive and has spread despite treatment with another androgen receptor pathway inhibitor (ARPI). This study is testing an investigational drug called AAA817. "Investigational" means it has not been approved by the United States Food and Drug Administration (FDA). In this study, participants will be randomly assigned (like flipping a coin) to receive AAA817 alone, AAA817 with an androgen receptor pathway inhibitor (ARPI), or standard of care treatment. The primary purpose of this study is to determine whether AAA817, given alone or in combination with an ARPI, is safe and effective compared to standard of care treatments. This drug is given to participants as a radioligand therapy infusion. Participants in this study can expect to be in the study for up to 6.1 years, including two visits before starting treatment, visits every 4 weeks during treatment, and visits every 12 weeks during long-term follow-up for up to 5 years after treatment ends.. There will be a total of 9 patients enrolled locally.

This study is for subjects that have been diagnosed with prostate cancer. Subjects can expect to remain in the study for up to 39 months or longer. There will be a total of 16 enrolled locally. The study is set to enroll subjects for 39 months.