A positive integer, $n$, is divided by $d$ and the quotient and remainder are $q$ and $r$ respectively. In addition $d$, $q$, and $r$ are consecutive positive integer terms in a geometric sequence, but not necessarily in that order.

For example, $58$ divided by $6$ has quotient $9$ and remainder $4$. It can also be seen that $4,6,9$ are consecutive terms in a geometric sequence (common ratio $3/2$).
We will call such numbers, $n$, progressive.

Some progressive numbers, such as $9$ and $10404={102}^2$, happen to also be perfect squares.
The sum of all progressive perfect squares below one hundred thousand is $124657$.

Find the sum of all progressive perfect squares below one trillion (${10}^{12}$).