If $x, y, z$ satisfy $x+y+z = 12$, $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 2$, and $x^3+y^3+z^3 = -480$, find $x^2 y + xy^2 + x^2 z + xz^2 + y^2 z + yz^2$.

Let's think step by step and output the final answer within \boxed{}.
user