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Usually if it backfires out the exhaust it's running rich, if it coughs out the intake it's running lean. You could also check the exhaust to see if their are any cracks or leaks anywhere. If you have the baffles out of the exhaust and have it rich enough to keep it from coughing thru the intake you probibly have it rich enough to backfire out the exhaust.
Question,what do you suggest,its not backfiring thru the carb,although its worth noting the idle screw had to be readjusted due to the fact that the spring which keeps it tight was broken and was replaced.
The condition you have is because of a lean condition due to your low speed jet. Go to the next size, then test ride. If this problem just started, and you did not add any new parts, you could have a clog in one of the circuts.