2025 IMC keystage3-IWYMIC team Question 4

For real $x$, let $[x]=\lfloor x\rfloor$ and $\{x\}=x-[x]$. Positive integers $(n,p,q)$ satisfy $q\gt p\gt3$, $n\le2025$ and $\{n/2\}+\{n/3\}+\{n/p\}+\{n/q\}=3$. Among triples minimizing $p+q$, how many possible values of $n$ are there?