Honestly, I use thermometers for reference only.
I check temps with an infrared temp-gun and then compare to thermometer readings.
Once the gun displays the temps I want I just keep an eye on the thermometers to ensure everything is working the way it should.
If budget is a concern I would advise a thermostat as the 1st purchase, and a temp gun as a second down the road.
109 is high, but, you need the temp where your snake is which is above the carpet. I expect you will soon get tired of carpet cleaning and want a different substrate.
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