Weird equality case (actually maybe not)

$\frac{\sqrt[3]{a^3+b^3} + \sqrt[3]{b^3+c^3} + \sqrt[3]{c^3 + a^3}}{ \left(\sqrt{a} + \sqrt{b} + \sqrt{c} \right)^2} \leq k$

Consider the sides $a,b,c$ of a triangle. Find the value of $\lfloor 100 \times k \rfloor$ where $k$ is the minimum positive real number such that the inequality above is fulfilled.