PMO 2023 Problem 7

A set of positive integers is said to be pilak if it can be partitioned into 2 disjoint subsets $F$ and $T$ , each with at least $2$ elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers. Find all positive integers $n$ such that the set containing all the positive divisors of $n$ except $n$ itself is pilak.