AMC 10 2022b Problem 24

Consider functions $f$ that satisfy $$|f(x)-f(y)|\leq \frac{1}{2}|x-y|$$ for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, what is the greatest possible value of $$f(f(800))-f(f(400))?$$ $\textbf{(A)}\ 25 \textbf{(B)}\ 50 \textbf{(C)}\ 100 \textbf{(D)}\ 150 \textbf{(E)}\ 200$