Nemotron-IMO-Bench 2026 N7

Let $(a_n)_{n\geq 1}$ be an increasing sequence of pairwise relatively prime positive integers and let $(b_n)_{n\geq 1}$ be a sequence of pairwise distinct positive integers. If for any $n\geq 0$ the number $\sum_{i=1}^n 1/b_i$ is an integer multiple of the number $\sum_{i=1}^n 1/a_i$, prove that $a_n=b_n$ for all $n$.