Alternating groups as products of four conjugacy classes

Let $G$ be the alternating group $Alt(n)$ on $n$ letters. We prove that for any $\epsilon >0$ there exists $N=N(\epsilon )\in \mathbb{N}$ such that whenever $n\ge N$ and $A$, $B$, $C$, $D$ are normal subsets of $G$ each of size at least $\mathrm{\mid }G{\mathrm{\mid }}^{1\mathrm{/}2+\epsilon }$, then $ABCD=G$.