Testing the Use of A Single Drug (Olaparib) or the Combination of Two Drugs (Cediranib and Olaparib) Compared to the Usual Chemotherapy for Women With Platinum Sensitive Ovarian, Fallopian Tube, or Primary Peritoneal Cancer

Inclusion Criteria: * Patients must have platinum-sensitive recurrent high-grade serous or high-grade endometrioid ovarian, primary peritoneal, or fallopian tube cancers; patients with other (clear cell, mixed epithelial, undifferentiated carcinoma, or transitional cell carcinoma) high-risk histologies are also eligible, provided that the patient has a known deleterious germline BRCA1 or BRCA2 mutation identified through testing at a clinical laboratory; Note: Due to the long acceptance of germline BRCA testing through Myriad, Myriad testing will be accepted; if testing for germline BRCA is done by other organizations, documentation from a qualified medical professional (e.g., ovarian cancer specialty physician involved in the field, high risk genetics physician, genetics counselor) listing the mutation and confirming that the laboratory results showed a recognized germline deleterious BRCA1 or BRCA2 mutation or BRCA rearrangement is required; please collect a copy of Myriad or other BRCA mutational analysis (positive or VUS or negative) reports * Platinum-sensitive disease defined as no clinical or radiographic evidence of disease recurrence for \> 6 months (or 182 days) after last receipt of platinum-based therapy * Patients must have had a complete clinical response to their prior line of platinum therapy and cannot have had progression through prior platinum-based therapy * Patients must have signed an approved informed consent and authorization permitting release o…