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Epicyclic motion and precession

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Imagine I am standing but not moving forward/backward, but I am rotating my body about my own axis. Now imagine if I moved in a circular path as I was spinning on my own axis. Is this considered to be precessing (assuming the axis of rotations are always in the z direction)?

My friend says it is, but I disagree. Wouldn't I always have the same angular momentum? And torque induced precession requires a change in L? Now I know there is a torque-free precession, but I didn't quite understand it. Can someone clear this up?

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