When does $\gcd(a^n+b^n,|a-b|)=\gcd(a+b,|a-b|)$?

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(I'll assume $a>b$ now)

As title, when does $\gcd(a^n+b^n,a-b)=\gcd(a+b,a-b)$?

I know that $\gcd(a+b,a-b)=$ either $\gcd(a, b)$ or $2\gcd(a, b)$, but I do not know how to proceed. It would be so much better if $a^n+b^n$ is replaced with $a^n-b^n$... haha. And no, this is not a homework question.

Thank you for the help