Nemotron-IMO-Bench 2026 A18

Find the largest real number $k$ such that for all positive real numbers $a,b,c,d$ with $ab+bc+cd+da=1$ $$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}-4\geq k(a^2+b^2+c^2+d^2-4).$$