Middle European Mathematical Olympiad 2023 Problem T-2

Let $a, b, c$ and $d$ be positive real numbers with $abcd = 1$. Prove that

$$\frac {a b + 1}{a + 1} + \frac {b c + 1}{b + 1} + \frac {c d + 1}{c + 1} + \frac {d a + 1}{d + 1} \geq 4,$$

and determine all quadruples $(a,b,c,d)$ for which equality holds.