Lets say a linear form $ax+by$ represents $n$ if $ax+by=n$ for some $x,y≥0$. 

Call a pair $a,b\in\Bbb N$ with $\mathsf{gcd}(a,b)=1$ a good coprime pair if
for some $r,s,t,u>1$ with each of $rs,tu,rt,su,ru,st<g(a,b)$ (frobenius number) $ax+by$ represents one pair among three pairs $(rs,tu)$, $(rt,su)$ and $(ru,st)$, then it should avoid at least one member of other two pairs.

Do such pairs exist at all?

If they do then 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 $[n,2n]$?

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A bad pair example:

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

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