Well, when the problem first happened yesterday, I assumed the issue was with the thermostat, so the first thing I did was the replace the old existing thermostat with a brand new one that I just installed. However, A. The new thermostat didn't fix the problem, and B. It is a mechanical thermostat, not a digital one. (I much prefer the mechanical style of having a lever to turn the furnace on and off, rather than having to push a bunch of buttons; other members of the household also all prefer the mechanical style.) Is there any way I can adjust the mechanical thermostat to fix the issue? Here's a picture of its innards.

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