BALKAN 2019 Problem 2

Let $a, b, c$ be real numbers, such that $0 \leq a \leq b \leq c$ and $a+b+c=a b+b c+c a>0$. Prove that $\sqrt{b c}(a+1) \geq 2$. Find all triples $(a, b, c)$ for which equality holds.