INMO 2016 Problem 2

For positive real numbers $a, b, c$, which of the following statements necessarily implies $a=b=c$ : (I) $a\left(b^{3}+c^{3}\right)=b\left(c^{3}+a^{3}\right)=c\left(a^{3}+b^{3}\right)$, (II) $a\left(a^{3}+b^{3}\right)=b\left(b^{3}+c^{3}\right)=c\left(c^{3}+a^{3}\right)$ ? Justify your answer.