Call a pair of integers $a,b\in\Bbb N$ with $\mathsf{gcd}(a,b)=1$ a good coprime pair if $$ax+by=rs\mbox{ and }ax'+by'=tu\mbox{ for some }r,s,t,u,x,y,x',y'>0$$$$\implies av+bw=ru\mbox{ and }av'+bw'=st\mbox{ satisfies }vw<0\mbox{ and }v'w'<0.$$

Is there an $n_0\in\Bbb N$ such that for every $n\in\Bbb N_{>n_0}$ contains a good coprime pair in $[n,2n]$?