2024 AMC 10A Problem 20

Let $S$ be a subset of $\{1, 2, 3, \dots, 2024\}$ such that the following two conditions hold:

• If $x$ and $y$ are distinct elements of $S$ , then $|x-y| > 2.$

• If $x$ and $y$ are distinct odd elements of $S$ , then $|x-y| > 6.$

What is the maximum possible number of elements in $S$ ?