On Thompson's conjecture for alternating and symmetric groups of degree greater than 1361

Let $G$ be a finite group $G$, and let $N(G)$ be the set of sizes of its conjugacy classes. It is shown that if $N(G)$ equals $N(\mathrm{Alt}_n)$ or $N(\mathrm{Sym}_n)$, where $n>1361$, then $G$ has a composition factor isomorphic to an alternating group $\mathrm{Alt}_m$ with $m\leq n$ and the half-interval $(m, n]$ contains no primes.