Final answer:
The first step is to provide immediate oxygen supplementation to improve the patient's oxygenation.
Step-by-step explanation:
Based on the given scenario, the oxygen delivery equipment has failed, and the 72-year-old female is experiencing cyanosis around her lips, fingers, and lower extremities. Cyanosis occurs when there is a lack of oxygen in the blood, resulting in a bluish discoloration of the skin. In this case, the patient's SpO2 is only 84%, which indicates a significantly low level of blood oxygen saturation. Therefore, the first step you should take is to provide immediate oxygen supplementation to improve her oxygenation.