A mind boggling question

Algebra Level pending

For positive real numbers \[a,b,c\] which of the following statements necessarily implies \[a = b = c\]

\[(I). a(b^{3}+c^{3})=b(a^{3}+c^{3})=c(a^{3}+b^{3})\] \[(II). a(a^{3}+b^{3})=b(b^{3}+c^{3})=c(c^{3}+a^{3})\]

Source:INMO(Indian National Mathematics Olympiad) 2016

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