An Algebraic Geometry Problem!

In a $\Delta ABC$ , let $a,\ b,\ c$ be the lengths of the sides of the triangle such that:

• $b^2=\dfrac{2a^4}{a^4+1}$
• $c^2=\dfrac{3b^6}{b^8+b^4+1}$
• $a^2=\dfrac{4c^8}{c^{12}+c^8+c^4+1}$

Find the value of $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}$ ?