TUYMAADA 2023 J6

An $\textit{Euclidean step}$ transforms a pair $(a, b)$ of positive integers, $a > b$ , to the pair $(b, r)$ , where $r$ is the remainder when a is divided by $b$ . Let us call the $\textit{complexity}$ of a pair $(a, b)$ the number of Euclidean steps needed to transform it to a pair of the form $(s, 0)$ . Prove that if $ad - bc = 1$ , then the complexities of $(a, b)$ and $(c, d)$ differ at most by $2$ .