A 45-year-old woman is admitted to the hospital with a swelling on the side of her face of 2 months' duration. Radiographic examination reveals a parotid gland tumor. An operative procedure is performed in which the tumor is removed from the parotid gland. Three months postoperatively the patient complains that her face sweats profusely when she tastes or smells food, and a diagnosis is made of Frey syndrome (gustatory sweating). Which of the following nerves was most likely injured during the procedure?