GAIC Math 2024 Problem 4

Let $n$ be a positive integer. An integer $k$ is called a "fan" of $n$ if and only if $0 \leqslant k \leqslant n-1$ and there exist integers $x$, $y$, $z \in \mathbf{Z}$ such that $x^2+y^2+z^2 \equiv 0 (\mathrm{mod} n)$ and $xyz \equiv k (\mathrm{mod} n)$. Let $f(n)$ denote the number of fans of $n$. Then f(2020) = \underline{\hspace{2cm}}.