IR-MO 2022 Problem 4

$a_1,a_2,\ldots$ is a sequence of nonzero integer numbers that for all $n\in\mathbb{N}$ , if $a_n=2^\alpha k$ such that $k$ is an odd integer and $\alpha$ is a nonnegative integer then: $a_{n+1}=2^\alpha-k$ . Prove that if this sequence is periodic, then for all $n\in\mathbb{N}$ we have: $a_{n+2}=a_n$ . (The sequence $a_1,a_2,\ldots$ is periodic iff there exists natural number $d$ that for all $n\in\mathbb{N}$ we have: $a_{n+d}=a_n$ )