I have another factoring problem from Gelfand's Algebra book that I've been unable to solve;

Problem 122. Factor:

(f) $\displaystyle (a-b)^3+(b-c)^3+(c-a)^3$

My try:

$\displaystyle (a-b)^3+(b-c)^3+(c-a)^3=((a-b)+(b-c))((a-b)^2-(a-b)(b-c)+(b-c)^2)+(c-a)^3=(a-c)(a^2-2ab+b^2-(ab-ac-b^2+bc)+b^2-2bc-c^2)+(c-a)^3=(a-c)(a^2-2ab+b^2-ab+ac+b^2-bc+b^2-2bc+c^2)-(a-c)^3=(a-c)(a^2-3ab+3b^2+ac-3bc+c^2)-(a-c)^3=?$

Thanks in advance.