 Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$. 
Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.


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