Putnam 2025 A1

Let $m_0$ and $n_0$ be distinct positive integers. For every positive integer $k$, define $m_k$ and $n_k$ to be the relatively prime positive integers such that $$\frac{m_k}{n_k} = \frac{2m_{k-1} + 1}{2n_{k-1}+1}.$$ Prove that $2m_k+1$ and $2n_k+1$ are relatively prime for all but finitely many positive integers $k$.