Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself such that for all integers $x$ and $y$, the difference $f(x) − f(y)$ divides $x^n − y^n$. The original answer involves parameters \varepsilon, d, and c. Provide the sum of all distinct possible values of \varepsilon + d + c.

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