Path connectivity from one lattice point to another

Question

Let matrix $A=\begin{bmatrix}1&1\\0&1\end{bmatrix}$. Given two vectors of nonnegative integers $X=\begin{bmatrix}x\\y\end{bmatrix}$ and $U=\begin{bmatrix}u\\v\end{bmatrix}$, where $u\ge x$ and $v\ge y$, $\gcd(x,y)=1$ and $\gcd(u,v)=1$, we would like to determine whether there exists a natural number $K$ and a finite sequence of matrices $(M_k)_{k=1}^K$, where $M_k\in\{A,\,A^T\},\,\forall k$ such that $U=\big(\prod_kM_k\big)X$. Is there a solution without evaluating all possible (finite number of) such matrix sequences?