Sequence of coprime numbers

Question

Fix a prime $p$ and positive integers $r,a,x,y$ such that $(x,y)=1$.

Prove (if the statement is true) that there exists a positive integer $s$ such that $$ x(s)=x+ap^r(p^s-1) $$ and $$ y(s)=y+ap^r(p^s-1)+p^{2r}(p^{2s}-1) $$ are coprimes