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The auger should not move with the belt removed, there is a brake up against the auger pulley that will stop you from tuning it.
If you can not slide you new belt up between the two pulleys then you may want to hinge your machine to make it easier. On either side of the auger housing there should be two bolts (each side) that keep your auger housing attached to the motor / drive housing. If you remove the tow upper bolts (one on each side) and loosen the two lower bolts, the machine should hinge on the lower bolts allowing you much more access to the belt system.