Middle European Mathematical Olympiad 2019 Problem T-8

Let $N$ be a positive integer such that the sum of the squares of all positive divisors of $N$ is equal to the product $N(N + 3)$. Prove that there exist two indices $i$ and $j$ such that $N = F_i \cdot F_j$, where $(F_n)_{n=1}^{\infty}$ is the Fibonacci sequence defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n \geq 3$.