Middle European Mathematical Olympiad 2019 Problem T-1

Determine the smallest and the greatest possible values of the expression $$\left(\frac{1}{a^2 + 1} + \frac{1}{b^2 + 1} + \frac{1}{c^2 + 1}\right)\left(\frac{a^2}{a^2 + 1} + \frac{b^2}{b^2 + 1} + \frac{c^2}{c^2 + 1}\right)$$ provided $a$, $b$, and $c$ are non-negative real numbers satisfying $ab + bc + ca = 1$.