KöMaL 2021 B5196

Let $p(x)=2x+1$. $A$ is a subset of set $S=\{1,2,\dots,2021\}$ such that it contains at most one of the numbers $n$, $p(n)$, $p\big(p(n)\big)$ for every $n$, but this condition will not hold anymore if any extra element of $S$ is added to $A$. What may be the number of elements in the set $A$?