Hello, It is a very common problem that the control heads on these vehicles have a internal bad connection on the circuit board. One simple test would be: when the power goes out on the control head smack the side of the dash board next to the climate control, if it comes back on you know you have a bad climate control head. I know it don't sound very professional, but it's a fast and accurate test.
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ASE Certified Technician
Over 25 years of experience in automotive computer systems. ASE L1 Cerfification