BALTIC-WAY 2020 Problem 2

Let $a, b, c$ be positive real numbers such that $a b c=1$. Prove that

$$\frac{1}{a \sqrt{c^{2}+1}}+\frac{1}{b \sqrt{a^{2}+1}}+\frac{1}{c \sqrt{b^{2}+1}}>2 .$$