An algebra problem by Shubham Haldēr

Suppose that $a, b$ and $c$ are non-zero numbers that satisfy

$\dfrac1a + \dfrac1b + \dfrac1c = \dfrac1{a+b+c}.$

If $n$ is an integer such that

$\dfrac1{a^n} + \dfrac1{b^n} + \dfrac1{c^n} = \dfrac1{a^n+b^n+c^n},$

then what is the best classification of $n$ ?