IMC 2017 Problem 5

Let $k$ and $n$ be positive integers with $n\geq k^2-3k+4$ , and let $$f(z)=z^{n-1}+c_{n-2}z^{n-2}+\dots+c_0$$ be a polynomial with complex coefficients such that $$c_0c_{n-2}=c_1c_{n-3}=\dots=c_{n-2}c_0=0$$ Prove that $f(z)$ and $z^n-1$ have at most $n-k$ common roots.