Have a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to trametinib, or excipients or to dimethyl sulfoxide (DMSO) or to either carboplatin or paclitaxel TREATMENT: Patients with a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to trametinib dimethyl sulfoxide (DMSO), its excipients, or DMSO, are ineligible to receive treatment with trametinib DMSO TREATMENT: Patients with a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to trametinib DMSO, its excipients, or DMSO, are ineligible to receive treatment with trametinib DMSO Known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to the study treatments, their excipients, and/or dimethyl sulfoxide (DMSO) Known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to the study treatments, their excipients, and/or dimethyl sulfoxide (DMSO) Have a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to trametinib or excipients or to dimethyl sulfoxide (DMSO) Have a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to trametinib, or excipients or to dimethyl sulfoxide (DMSO), or to compounds of similar chemical or biologic composition to navitoclax History of known immediate or delayed hypersensitivity reactions attributed to compounds of similar chemical or biologic composition to AT13387, dabrafenib, or trametinib, or excipients or to dimethyl sulfoxide (DMSO) Known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to trametinib, or excipients, or to dimethyl sulfoxide (DMSO), or to Cremophor EL (polyoxyethylated castor oil); please note, exclusion for Cremophor is unnecessary unless paclitaxel is the only agent available and the patient randomizes to the conventional therapy option Have a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to trametinib, or excipients or to dimethyl sulfoxide (DMSO) or to thyrotropin alpha (Thyrogen) Known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to the study treatments, their excipients, and/or dimethyl sulfoxide (DMSO) CLINICAL/LABORATORY CRITERIA: Patients must not have an immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to trametinib, or excipients, or to dimethyl sulfoxide (DMSO) or other agents used in the study Have a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to trametinib, or excipients or to dimethyl sulfoxide (DMSO)