Given small $\epsilon>0$ how small should $n\in\Bbb N$ be such that if $a,b,c,d,q,r,u,v,x,y,m,m'\in\Bbb N$ with $gcd(a,b)=gcd(a,x)=gcd(b,y)=1$ the following relations can hold with constraints $c,d=(n^3,2n^3)$, $a,b\in(n,2n)$, $x\in(1,a)$, $y\in(1,b)$, $m,m'=\theta(n)$ ($\theta$ is landau function), $u,v\in(n^{2+\epsilon},2n^{2+\epsilon})$ and $q,r\in(n^4,2n^4)$? $$c{b^2}=au + qm,\quad q=ca-x$$ $$d{a^2}=bv + rm',\quad r=db-y$$
In other words given small $\epsilon>0$ and large $n\in\Bbb N$ how many $a,b,c,d,q,r,u,v,x,y,m,m'\in\Bbb N$ satisfy these constraints?
Note these imply $$c^2b^2-cqm=qu+xu$$ $$d^2a^2-drm'=rv+yv$$