Carboplatin-Enhanced Therapy in BRCA-Mutated or Neuroendocrine-Differentiated Aggressive Metastatic Castration-Sensitive Prostate Cancer

Inclusion Criteria: 1. Histologically or cytologically proven adenocarcinoma of the prostate (any T stage, Gleason score or PSA level). 2. De novo metastatic disease documented by a positive bone scan or CT scan or an MRI. For patients with nodal metastases only, only patients with extra-pelvic enlarged lymph nodes (lymph nodes located above the iliac bifurcation) can be included if they have either: at least one extra-pelvic lymph node ≥ 2 cm, or extra-pelvic lymph node(s)≥1 cm if the patients also have at least one pelvic lymph node ≥ 1.5 cm. 3. Presence of pathogenic BRCA gene alteration (BRCAm cohort) OR histologically adenocarcinoma of the prostate with neuroendocrine differentiation (NED cohort) 1. For the BRCAm cohort: confirmation of a pathogenic BRCA gene alteration by historical analysis, using one of the following: * Circulating tumor DNA (ctDNA) analysis * Tumor tissue analysis (archival sample permitted) * Historical germline testing results 2. For the NED cohort: histologically adenocarcinoma of the prostate with neuroendocrine differentiation41. NED is defined as either: * Presence of mixed tumors with prostate adenocarcinoma associated with small cell or large cell neuroendocrine carcinoma according to WHO classification55 (5th edition). Pure small cell or large cell neuroendocrine carcinoma is not accepted (prostate adenocarcinoma component should express NKX3.1, AR and/or PSA to exclude pure small cell or large cell NE carcinoma) * Or immunohistochemical staining showing some degree of neuroendocrine differentiation within adenocarcinoma (positivity of at least 2 of the following: chromogranin A, synaptophysin, TTF1 or INSM1). Threshold for NE markers positivity is ≥ 1% tumor cells. Co-expression of neuroendocrine markers with AR, NKX3.1 and/or PSA is allowed for these cases. If the patient has both a BRCA alteration and evidence of NED on his biopsy, he will be randomized in the BRCA cohort. 4. Patients with ECOG PS ≤ 1, or patient with PS 2, provided PS alteration is disease-related. 5. Life expectancy of at least 6 months. 6. Male aged ≥ 18 years old and ≤ 80 years old. 7. Hematology values: Hemoglobin ≥ 10.0 g/dL, Platelet count ≥ 100,000/mL, Neutrophil ≥ 1500 cells/mm³ (or neutrophil ≥ 1000 cells/mm³ in case of ethnic neutropenia). 8. Biochemistry values: Renal function: Serum creatinine \< 1.5 x ULN or a calculated creatinine clearance ≥ 60 mL/min, Liver function: Serum bilirubin ≤ 1.5 x ULN (except for patients with documented Gilbert's disease), AST and ALT ≤ 1.5 x ULN (and ≤ 5 ULN in case of liver metastases), ALK-P ≤ 2.5 x ULN (in case of bone metastasis, ALK-P\<1000U/L if bilirubin is normal). 9. Patients must have received ADT and ARPI (abiraterone, enzalutamide, apalutamide, darolutamide) for a maximum of 3 months before randomization. Prior ARPI in mHSPC is allowed if required prior to randomization, with no radiographic evidence of disease progression or rising PSA levels prior to Day 1. 10. Patients willing and clinically fit to receive docetaxel with no contraindication to docetaxel according to the Summary of Product Characteristics (SmPC) of the drug. 11. Patients able to take oral medication. 12. Patients who have received the information sheet and signed the informed consent form 13. Patients must be affiliated to a social security system or beneficiary of the same. 14. Male patients who have partners of childbearing potential and/or pregnant partners must use a method of birth control in addition to an adequate barrier protection (condoms) as determined to be acceptable by the study doctor during the treatment period and for 6 months after the last dose of Docetaxel. Exclusion criteria: 1. Patients with previous definitive local treatment directed to prostate primary cancer (radiotherapy, brachytherapy, radical prostatectomy, ultrasound, cryotherapy, or other). A previous trans-urethral resection of the prostate (TURP) and previous local treatments of metastases are allowed. 2. Prior cytotoxic chemotherapy or biological therapy for the treatment of prostate cancer. Prior exposure to carboplatin or other platinum containing compounds. 3. Nervous system disorder (paresthesia, peripheral motor neuropathy, or peripheral sensory neuropath\\\>= grade 2, per the Common Terminology Criteria for Adverse Events version 6.0. 4. Patients known to be human immunodeficiency virus (HIV) positive, symptomatic viral hepatitis or chronic liver disease. 5. Patient with administration of live or live attenuated vaccines (strongly discouraged in patients) and is formally contraindicated in the case of the yellow fever vaccine. 6. Severe or moderate hepatic impairment (Child - Pugh class C or B). 7. Clinically known significant heart disease in the past 6 months as evidenced by myocardial infarction, or arterial thrombotic events, severe or unstable angina, or New York Heart association (NYHA) Class II-IV heart disease or cardiac ejection fraction measurement of \< 50% at baseline, atrial fibrillation, or other cardiac arrhythmia requiring therapy. 8. History of malignancy, except non-melanoma skin cancer, with low risk of recurrence within 24 months. 9. Known allergies, hypersensitivity or intolerance to carboplatin or other platinum containing compounds. 10. Pathological finding consistent with pure small cell carcinoma of the prostate. 11. Patients already included in another therapeutic trial involving an experimental drug (patient in a non-experimental trial with no modification of the patient's care can be included). 12. Patients with significantly altered mental status prohibiting the understanding of the study or with psychological, familial, sociological or geographical condition potentially hampering compliance with the study protocol and follow-up schedule or any condition which, in the opinion of the investigator, would preclude participation in this trial. 13. Patient under guardianship or deprived of his liberty by a judicial or administrative decision or incapable of giving its consent.