Because $$\frac{2ax}{ab+a+1} = \frac{2ax}{ab+a+abc}=\frac{2x}{b+1+bc}$$ $$\frac{2cx}{ca+c+1} = \frac{2bcx}{cab+bc+b}=\frac{2bcx}{1+bc+b}$$ \begin{align*} \implies 1 &= \frac{2ax}{ab+a+1}+\frac{2bx}{bc+b+1}+\frac{2cx}{ca+c+1} \\ &= \frac{2x}{b+1+bc}+ \frac{2bx}{bc+b+1}+\frac{2bcx}{1+bc+b}\\ &= \frac{2x(1+b+bc)}{bc+b+1}\\ &= 2x \end{align*} $$\therefore\quad x=\boxed{\frac{1}{2}}$$