The same procedure was used in class to explain the motion of the axis of gyroscope. This problem also is for the practice of the 'right-hand rule' of the cross product. Go over the lecture note carefully first. In the class, the rotation around the vertical axis is slow and can be neglected when calculating the angular kmomentum. Recall that the total angular momentum is given by the angular momentum of c.m. motion.
L cm where L cm is the angular momentum with respect to the c.m.
Actually it is close but not quite.