INMO 2024 Problem 4

A finite set $S$ of positive integers is called cardinal if $S$ contains the integer $|S|$, where $|S|$ denotes the number of distinct elements in $S$. Let $f$ be a function from the set of positive integers to itself, such that for any cardinal set $S$, the set $f(S)$ is also cardinal. Here $f(S)$ denotes the set of all integers that can be expressed as $f(a)$ for some $a$ in $S$. Find all possible values of $f(2024)$.

Note: As an example, $\{1,3,5\}$ is a cardinal set because it has exactly 3 distinct elements, and the set contains 3 .