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

$g(a,b)$ is Frobenius number.

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

A bad pair example:

$a=22,b=21,s = 16, t = 17,r = 19,u = 15$

$$10a+4b=rs$$ $$8a+7b=rt$$ $$9a+2b=su$$ $$3a+9b=tu$$