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

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

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A bad pair example:

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

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