BALTIC-WAY 2022 Problem 1

Let $\mathbb{R^+}$ denote the set of positive real numbers. Assume that $f:\mathbb{R^+} \to \mathbb{R^+}$ is a function satisfying the equations: $$f(x^3)=f(x)^3 \text{and} f(2x)=f(x)$$ for all $x \in \mathbb{R^+}$ . Find all possible values of $f(\sqrt[2022]{2})$ .