Existence of a cycle of length $2^k$?

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I was given a question which asks me to show that if each vertex of a graph has degree$\geq 3$, then it has a cycle whose length is some power of $2$. I have been able to show that it has a even cycle.it can be shown either by induction or by considering a maximal path. but I have not been able to progress beyond that. can some one help me?