Least Squares with Singular $AA^T$

Question

Given the following system, find all least squares solutions: $\begin{bmatrix}1 & 2 & 3\\2 & 3 & 4\\3 & 4 & 5\end{bmatrix} \vec{x} = \begin{bmatrix}1\\1\\2\end{bmatrix}$ However, after trying to minimize residuals with: $\vec{x} = (A^TA)^{-1}A^T\begin{bmatrix}1\\1\\2\end{bmatrix}$ I found that $det(AA^T)$ is singular... I think this means that their exist infinitely many least squares solutions to the system, but I don't know how to go about describing them all. I am relatively new to linear algebra (Uni level into class at the moment) so any help/explanation would be great!