An algebra problem by Mehul Chaturvedi

Algebra Level 4

Let $f(x)=x^3+ax^2+bx+c$ and $g(x)=x^3+bx^2+cx+a,$ where $a,b,c$ are integers with $c\neq0.$ Suppose that the following conditions hold:

(i) $f(1)=0;$

(ii) the roots of $g(x)=0$ are the squares of the roots of $f(x)=0.$

Find the value of $a^{2013}+b^{2013}+c^{2013}.$