There are two study questions we are asking in this randomized phase II/III trial based on a blood biomarker, Epstein Barr virus (EBV) deoxyribonucleic acid (DNA) for locoregionally advanced non-metastatic nasopharyngeal cancer. All patients will first undergo standard concurrent chemotherapy and radiation therapy. When this standard treatment is completed, if there is no detectable EBV DNA in their plasma, then patients are randomized to either standard adjuvant cisplatin and fluorouracil chemotherapy or observation. If there is still detectable levels of plasma EBV DNA, patients will be randomized to standard cisplatin and fluorouracil chemotherapy versus gemcitabine and paclitaxel. Radiation therapy uses high energy x rays to kill tumor cells. Drugs used in chemotherapy, such as cisplatin, fluorouracil, gemcitabine hydrochloride, and paclitaxel work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. It is not yet known whether giving cisplatin and fluorouracil is more effective than gemcitabine hydrochloride and paclitaxel after radiation therapy in treating patients with nasopharyngeal cancer.