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This is going to take some troubleshooting with a multi-meter. Do you have one?
Then there isn't much you can do to troubleshoot this. About the only thing you can do is bypass the thermostat (in case that's the culprit) by connecting the R & W terminals on the control board. the board in the the control box in the blower compartment. There are also rollout switches (see attached pic for typical) just above and outside the burner chamber and on the draft hood. You push the button on the switch. If it clicks you have reset it.