If you are found to be eligible to take part in this study, you will be assigned to a dose level of the study drug combination that is based on when you joined this study. Up to 6 dose levels of the study drug combination will be tested. Three (3) participants will be enrolled at each dose level. If no intolerable side effects occur in the first group of 3 people, the next 3 people enrolled in the study will receive the same dose of carboplatin and a higher dose of TAS-106. If this second group has no intolerable side effects, the next 3 people will receive a higher dose of carboplatin while receiving the same dose of TAS-106 as the second group. This will continue until the highest safe dose combination is found.