HMMT 2022 T4

Suppose $n \geq 3$ is a positive integer. Let $a_{1}<a_{2}<\cdots<a_{n}$ be an increasing sequence of positive real numbers, and let $a_{n+1}=a_{1}$. Prove that

$$\sum_{k=1}^{n} \frac{a_{k}}{a_{k+1}}>\sum_{k=1}^{n} \frac{a_{k+1}}{a_{k}} .$$