Determine all positive integers $n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y,z) = (x - y)^{5n} + (y - z)^{5n} + (z - x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients.

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