### Instruction: Suppose that there exist nonzero complex numbers $a,$ $b,$ $c,$ and $d$ such that $k$ is a root of both the equations $ax^3 + bx^2 + cx + d = 0$ and $bx^3 + cx^2 + dx + a = 0.$  Enter all possible values of $k,$ separated by commas.### Response: I notice that the two equations have the same coefficients, but shifted by one term.
This suggests that there might be some symmetry or pattern in the roots.
I also notice that if I multiply the first equation by $x$ and subtract the second equation, I get $a(x^4 - 1) = 0.$
This means that either $a = 0$ or $x^4 = 1.$
But since $a$ is nonzero by assumption, I can conclude that $x^4 = 1.$
This implies that $x$ is a fourth root of unity, which are complex numbers of the form $e^{2\pi i n / 4}$ for $n = 0, 1, 2, 3.$
So, the possible values of $k$ are $e^{2\pi i n / 4}$ for $n = 0, 1, 2, 3,$ or in simpler terms, $1, i, -1, -i.$