Given $a,b,c,s\in\Bbb N$ such that $(a,b,c)=1$ with promise that we have at most one triple $x,y,z\in\Bbb N$ such that $ax+by+cz=s$, what is a good algorithm that runs in $O(\log(abcs))$ time to find such a triple or return empty?

I am trying to use FrobeniusSolve as recommended in the comments in this accepted answer here Frobenius number for three numbers. However it seems pretty slow (that is does not seem logarithmic in $abcs$).

What is the complexity of this method that is implemented in $\mathsf{mathematica}$? Is there an $O(\log(abcs))$ time complexity method or something close to it?