KZ-MO 2017 Problem 3

$\{a_n\}$ is an infinite, strictly increasing sequence of positive integers and $a_{a_n}\leq a_n+a_{n+3}$ for all $n\geq 1$ . Prove that, there are infinitely many triples $(k,l,m)$ of positive integers such that $k<l<m$ and $a_k+a_m=2a_l$