Weird equality case (actually maybe not)

Algebra Level 5

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

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.

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