ISL 2023 N5

Let $a_{1}<a_{2}<a_{3}<\cdots$ be positive integers such that $a_{k+1}$ divides $2\left(a_{1}+a_{2}+\cdots+a_{k}\right)$ for every $k \geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_{k}$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_{k}$. (Netherlands)