Tessellate S.T.E.M.S - Mathematics - College - Set 3 - Problem 3

Let $p(x)$ be a polynomial of degree $n$ with complex coefficients, with $k$ distict complex roots. For a particular complex-valued sequence $\{\displaystyle a_n \}_{n \geq 1}$ , the sequence $\displaystyle \{ p(a_n)\}_{n \geq 1}$ converges to $0$ . Consider a convergent subsequence $\{b_n\}$ of $\{a_n\}$ , define $\displaystyle L := \lim_{n\to \infty} b_n$

How many distinct values can $L$ have?

This problem is a part of Tessellate S.T.E.M.S.