Nemotron-IMO-Bench 2026 A39

Let $a,b,c,d$ be positive real numbers such that $ab+bc+cd+da=4$ and at most one of $a,b,c,d$ is smaller than $1$. Prove that $$\frac{1}{ab+3}+\frac{1}{bc+3}+\frac{1}{cd+3}+\frac{1}{da+3}+\frac{1}{ac+3}+\frac{1}{bd+3}\geq \frac{3}{2}.$$