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This may be a flame sensor issue.
The flame sensor rod is a small thin steel rod that is straight or L shaped and is usually located near the pilot igniter or in the path of the burners.
Locate it, remove it, and clean it with a piece of emery cloth or sandpaper.
Scrub it down very good.
Also make sure you have a clean air filter.
Is there any possibility that the sensor at the end of the flue box could be faulty?
Is this sensor opening up?
All I can say is that the relay next to the transformer is what kicks off during the cycles.
Try cleaning the flame sensor then see what happens.
Let me know what happens.
I've cleaned the flame sensor but the cycling continues.
Have you changed the filter?
I clean the filter once a month. It's a washable type.
It actually sounds to me as if the furnace is overheating and the high limit switch is shutting it down.
Do you feel that this is possible?
It sounds about right to me
Does this system have a a/c system?
I am going to opt out so another expert can look at this.
I am sorry that I could not resolve the issue.