MEMO 2023 T2

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

$$\frac{ab + 1}{a + 1} + \frac{bc + 1}{b + 1} + \frac{cd + 1}{c + 1} + \frac{da + 1}{d + 1} \geq 4,$$

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