GAIC Math 2024 Problem 11

Define the function $f(x) =[x[x]]$, where $[x]$ represents the largest integer not exceeding $x$. For example, $[-2.5] = -3$.For a positive integer $n$, let $a_n$ be the number of elements in the range set of $f(x)$ when $x \in [0,n)$. Then the minimum value of $\frac{a_n + 90}{n}$ is \underline{\hspace{2cm}}.