Compute reduced word for element in particular free group

Question

Let's consider some particular free group, for instance, the example $G = \langle A, B \rangle < \text{SL}(2, \mathbb{Z})$ in Wikipedia about ping-pong lemma, and some element $g \in G$.

Problem #1 is to compute the reduced word for this $g$, i.e. the corresponding sequence of factors $A$, $A^{-1}$, $B$, $B^{-1}$ (maybe it's the simple problem, but I can't figure out anything better then iterating over possible sequences).

Problem #2 assumes that we have also some $x$, for example $x \in \mathbb{R}^2$, and the action $g. x \in \mathbb{R}^2$, and we need to find reduced word for $g$ using this information (again, maybe it's easy to reduce problem 2 to problem 1, but I'm confused with this a little).

P.S. The application of this will be the following: given the $x, y \in \mathbb{R}^2$ find the $g \in G$ that might give $y$ by acting on $x$. It's like "rounding" to the nearest group element in embedding space $\mathcal{H}$ if we already embedded $G$ in $\mathcal{H}$, for example, using Švarz–Milnor lemma. So the natural requirement for the procedures in answers on problems 1 and 2 is their computational efficiency, because this "rounding" smust be performed as fast as possible.

Answer 1

See the book, Kargapolov-Merzljakov "Fundamentals of group theory", section 14, "Free group". That group is called the "Sanov subgroup" there. The subgroup consists of all $2\times 2$ integral matrices $X$ with $\det(X)=1$, $x_{11}\equiv x_{22}\equiv 1 \pmod 4$, $x_{12}\equiv x_{21}\equiv 0\pmod 2$. On page 94 they give an algorithm to represent such a matrix as a product of generators $e_{12}(2), e_{21}(2)$.That representation is unique because the subgroup is free.