Powers of subsets in free periodic groups

It is proved that for every odd $n \ge 1039$ there are two words $u(x, y), v(x,y)$ of length $\le 2^{22}n^3$ over the group alphabet $\{x,y\}$ of the free Burnside group $B(2 ,n),$ which generate a free Burnside subgroup of the group $B(2,n)$. This implies that for any finite subset $S$ of the group $B(m,n)$ the inequality $|S^t|>4\cdot 2.9^{[\frac{t}{2^{22}s^3}]}$ holds, where $s$ is the smallest odd divisor of $n$ that satisfies the inequality $s \ge 1039$.