Selumetinib and Akt Inhibitor MK-2206 in Treating Patients With Refractory or Advanced Gallbladder or Bile Duct Cancer That Cannot Be Removed By Surgery

Inclusion Criteria: * Patients must have surgically unresectable histologically confirmed biliary tract adenocarcinoma (defined as gallbladder cancer, extrahepatic and intrahepatic cholangiocarcinoma; this definition excludes ampullary cancers and all tumors of mixed histology); cytological confirmation is not allowed on this study, as tissue is needed for correlative science; fresh tissue (mandatory) AND paraffin embedded tissue (positron emission tomography \[PET\]) from tumor blocks (if available) will be required from patients before enrolling on this study * Patients will be required to undergo a biopsy prior to enrolling on the study and will be given the option to have another biopsy around 4 weeks from initiation of treatment * Patients must have measurable disease by RECIST 1.1 criteria, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \>= 10 mm with spiral computed tomography (CT) scan (CT scan slice thickness no greater than 5 mm); malignant lymph nodes will be considered measurable if they are \>= 15 mm in short axis * All of the following: * Patients must have received only one prior line of systemic therapy for recurrent or advanced disease * Prior adjuvant therapy (chemotherapy +/- radiation) completed within 6 months of diagnosis of recurrence/metastases is equivalent to one line of prior therapy for metastatic disease * For patients who completed adjuvant therapy \> 6 months …