Prove $x^2+4y^2<1$ given $x^3 + y^3 = x-y$ and x and y are positive real numbers. (Using AM-GM)

Question

I have tried the following and other methods, however I haven't been able to solve this. I would very much appreciate if someone can point me in the right direction on how this can be solved.
$x-y=(x+y)(x^2 -xy+y^2)$
$(x+y)((x-y)^2-xy) = (x-y)$
$(x-y)[(x-y)(x+y) -1] - xy(x+y) = 0$
However this does not seem to lead anywhere.
This question is supposed to be solved with AM-GM however I haven't been able to apply it so far in any way which will lead to the inequality.