Search to the Fire
It's a fact of life that most departments only have the manpower to choose between our two most high-priority assignments - Search and Rescue or Fire Attack. Look at any Command Post tactical board or handheld tactical worksheet/job aid, and you'll see those two priorities front and center.
For those of us that don't have enough first-arriving firefighters to do both at the same time, there's an answer. It is a compromise that may not completely accomplish both primary goals, but it's realistic for the average house fire. That assignment is "Search to the Fire'.
For a 1st-due engine company, that assignment means to do what usually happens anyway - the engine will select a line, take it inside, and drag it through a primary access-egress route until they find the seat of the fire and hit it. If, on the way to the fire, they find a victim or victims, they make the rescue, protecting themselves and the victim with the hoseline if they can.
If there is no EMS on the scene and the rescue is successful, the pump operator can provide basic patient care (oxygen, first aid) while the engine crew re-enters to do further Search to the Fire.
Engine companies are limited to primary access and egress routes most of the time - dragging hose is going to limit your route of travel. If you don't have a truck company or other unit close by to do the search without a hoseline, then you're going to be limited to the primary access route, anyway.
Most command post boards and tactical worksheets don't have Search to the Fire as an option. Maybe it's time that we recognized what the 1st due engine actually does a lot of the time and start calling it what it actually is.
If we assign the 1st engine to Primary Search, then they're going to be slowed if they drag a hoseline and unable to fight any real fire without it. If we assign them to Fire Attack, they sometimes get "Candlemoth" syndrome and go straight to the flames without looking around for victims that might otherwise be found.
Search to the Fire tools for a two-firefighter team include the following
Charged hoseline with appropriate nozzle
Married Irons
Thermal Imager
Personal tools (door chocks, lights, wire cutters, screwdrivers, etc.)
If you go heavier than that, you may have so much stuff that you'll be ineffective at both the search and fire attack assignments.