Nemotron-IMO-Bench 2026 A3

Prove that for all positive real numbers $a,b,c,d$ $$\frac{a+b}{1+c}\cdot \frac{b+c}{1+d}\cdot \frac{c+d}{1+a}\cdot \frac{d+a}{1+b}\geq \frac{16abcd}{(1+\sqrt[4]{abcd})^4}.$$