VMO 2023 Problem 3

Find the maximum value of the positive real number $k$ such that the inequality $$\frac{1}{kab+c^2} +\frac{1} {kbc+a^2} +\frac{1} {kca+b^2} \geq \frac{k+3}{a^2+b^2+c^2}$$ holds for all positive real numbers $a,b,c$ such that $a^2+b^2+c^2=2(ab+bc+ca).$