Can there be a planet for which orbit and precession take the same time?

Is there a necessary relation between orbital period and the time of precession? In the case of Earth, the orbital period is one year, while a complete precession cycle takes thousands of years. Is a planet that completes a precession cycle within one of its "years" possible? Besides one of the poles having perpetual night and the other perpetual day, what else would this entail?

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Can there be a planet for which orbit and precession take the same time?

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