Let $$a,b,c$$ be reals satisfying $$3ab+2=6b$$, $$3bc+2=5c$$, and $$3ca+2=4a$$. Let $$\mathbb{Q}$$ denote the set of all rational numbers. Given that the product $$abc$$ can take two values $$\frac{r}{s}\in\mathbb{Q}$$ and $$\frac{t}{u}\in\mathbb{Q}$$, in lowest form, find $$r+s+t+u$$.