The technician should first check the thermal paste because it facilitates heat transfer from the CPU to the heatsink. If the thermal paste is old, improperly applied, or missing, the CPU may overheat, leading to sudden shutdowns after prolonged use. The question states the internal components are free of dust and debris, so cleaning does not making sense as a next step. Checking the ambient room temperature is important for overall heat management, but the first point of heat transfer that should be checked is between the CPU and heatsink. BIOS firmware updates could resolve overheating only if the issue is related to improper fan control; this step is typically considered after verifying the physical heat transfer mechanisms.