Can we solve the ECC DLP if we can distinguish whether the doubling of a public key is accompanied by reduction (modulo n) or not?

Let $E$ be an elliptic curve over a prime or a binary extension field $GF(2^m)$, and let $G(x_g,y_g)$ be a generator point on the curve. Let $Q$ be an arbitrary point $Q = r*G$, with $r$ scalar, and $Q$ an element from the group of generator $G$ of order $n$.

I have read in some sources (e.g. here for curves over binary extension fields) that, if an actor can distinguish whether the doubling of $Q$ is accompanied by reduction (modulo $n$), then it mathematically follows that he/she can distiguish between utilizing the algorithm of division (0) or subtraction-division to reverse the sought-for number $2^l G$ or $(2^l + 1) G$, which requires no more than $log_2n$ divisions and thus reverse the elliptic curve multiplication and solve the DLP for binary elliptic curves.

Yet, I do not follow why knowledge of whether a doubling is reduced mod $n$ or not provides enough information to solve the DLP. Can someone elaborate?