I know the following facts: $\text{SL}_2(\mathbb{Z})$ is generated by everyone's favorite matrices \begin{equation*} S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \end{equation*} and \begin{equation*} T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \end{equation*} and $\text{SL}_2(\mathbb{Z})$ is acts transitively on $\mathbb{P}^1(\mathbb{Q})$.

I have been told the answer to the following questions have something to do with continued fraction expansions, but I would like to find a reference/pointers to the particulars.

(1) How do I write a given matrix $A \in \text{SL}_2(\mathbb{Z})$ in terms of $S$ and $T$?

(2) For a given $p/q \in \mathbb{Q}$, how do I find an element $A \in \text{SL}_2(\mathbb{Z})$ with $A(\infty) = p/q$?

I know the following facts: $\text{SL}_2(\mathbb{Z})$ is generated by everyone's favorite matrices \begin{equation*} S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \end{equation*} and \begin{equation*} T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \end{equation*} and $\text{SL}_2(\mathbb{Z})$ acts transitively on $\mathbb{P}^1(\mathbb{Q})$.

I have been told that the answer to the following questions have something to do with continued fraction expansions, but I would like to find a reference/pointers to the particulars.

(1) How do I write a given matrix $A \in \text{SL}_2(\mathbb{Z})$ in terms of $S$ and $T$?

(2) For a given $p/q \in \mathbb{Q}$, how do I find an element $A \in \text{SL}_2(\mathbb{Z})$ with $A(\infty) = p/q$?