Matrices & Systems · Hard problem

A forwarding app relays a message count according to $T = \begin{pmatrix}2 & 1\\1 & 2\end{pmatrix}$ every round: if the current pair of counts is $\mathbf x$, the next round's pair is $T\mathbf x$. After exactly TWO rounds (that is, after applying $T$ twice, giving $T^2\mathbf x_0$), the pair of counts is $\begin{pmatrix}23\\22\end{pmatrix}$. Find the first entry of the original pair $\mathbf x_0$.