BALTIC-WAY 2021 Problem 20

Let $n \geq 2$ be an integer. Given numbers $a_{1}, a_{2}, \ldots, a_{n} \in\{1,2,3, \ldots, 2 n\}$ such that $\operatorname{lcm}\left(a_{i}, a_{j}\right)>2 n$ for all $1 \leq i<j \leq n$, prove that

$$a_{1} a_{2} \cdots a_{n} \mid(n+1)(n+2) \cdots(2 n-1)(2 n) .$$