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Let's try this: Leave the thermostat calling for heat, then go to the furnace and unplug it then plug it back in. Now, watch the stages carefully and tell me what it does.
It should start the induced draft motor (small motor in the burner compartment), igniter should spark or glow orange, burners should fire, blower should come on.
Okay, I see now.
Most likely the system is going into lockout. Next time this happens, look through the little quarter sized sight glass in the bottom panel. There will be an LED flashing a code. Count how many flashes and let us know.
Okay, 1 flash indicates an ignition failure lockout (tries 3 times than locks out). It is strange that it will start and heat the house after you reset it, but it could be that the flame sensor isn't sensing flame every time.
First, lets try cleaning the flame sensor with steel wool to eliminate this as a possibility. See picture below: