Alternating groups as products of four conjugacy classes

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

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