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x2 what Bill wrote, although I'd think the GFCI outlet is OK if nothing else trips it. There really isn't all that much to a furnace - just a blower & associated safety stuff that keeps it on when there's heat in the exchanger. Possibly there's something going on with the thermostat that would cause this (but that's also DC)? You've got a weird one for certain and may have to take the furnace & its wiring apart.
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