BG-JBMO-TST 2022 Problem 2

Let $a$ , $b$ and $c$ be positive real numbers with $abc = 1$ . Determine the minimum possible value of $$\left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \cdot \left(\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}\right)$$ as well as all triples $(a,b,c)$ which attain the minimum.