MathNet 2013 Question 82

Let $a$, $b$, $c$, $d \gt 0$ satisfying $abcd = 1$. Prove that
$$
\frac{1}{a+b+2} + \frac{1}{b+c+2} + \frac{1}{c+d+2} + \frac{1}{d+a+2} \le 1.
$$