Given large $t$ are there integers $t<A,B,m,n,m',n'<2t<t^2/2<|n'A-mB|,|-m'A+nB|$ with $mn'-m'n=AB$, $m+m'$ $=n+n'=A+B$ and $gcd(A,B)=$ $gcd(n'A-mB,-m'A+nB)= gcd((n'A-mB)(-m'A+nB),(A+B))=\pm1$?
$mn'-m'n=AB$, $m+m'=n+n'=A+B\implies m(A+B-n)-(A+B-m)n=AB$ $\implies (m-n)(A+B)=AB$.