Final answer:
The most important action for the nurse to take is to notify the patient's physician.
Step-by-step explanation:
The most important action for the nurse to take when a patient with septicemia develops prolonged bleeding from venipuncture sites and blood in the stools is to notify the patient's physician. Septicemia, also known as sepsis, is a severe infection that can lead to life-threatening complications. Bleeding from venipuncture sites and blood in the stools are signs of systemic infection and may indicate worsening septicemia. The physician needs to be informed immediately so that appropriate interventions can be initiated, such as adjusting antibiotic treatment or providing additional supportive care.