Waring number for any word in the alternating group is at most two

Statement
Suppose $$G$$ is the alternating group of degree $$n$$ for any $$n \ge 3$$ and $$w$$ is any word. Then, the Waring number for $$w$$ in $$G$$ (in the monoidal sense, and hence also in the group sense) is at most two.