Trying to find the set of unique representatives for the geodesics in the group $\langle a,t \mid ata^{-2}t^2a^{-2}tat^{-4}\rangle$

Question

I am studying the conjugacy growth of the groups, and I encountered the following group: $$G=\langle a,t \mid ata^{-2}t^2a^{-2}tat^{-4}\rangle$$

Thanks to Derek for pointing out that $G$ is an infinite hyperbolic group.

Question: Given a word in $\{a^{\pm 1}, t^{\pm 1}\}$, is there a way to check whether this word is a geodesic or not? (Or are there certain criteria for a word to be a geodesic)

Any ideas/references would be really appreciated.