= Solution
We induct on the <nilpotency class> $s$. For $s=1$, the group is <Abelian>, so take $C_1=A$ and no $X_i$. For $s>1$, part (i) writes
$$
A\subseteq XB_1\cdots B_k
$$
with $X$ small and every $\langle B_i\rangle$ of class at most $s-1$. Apply the induction hypothesis to each $B_i$. Since $B_i\subseteq A^{O(1)}$ and its approximation parameter is $K^{O(1)}$, all resulting small sets lie in $A^{O_s(1)}$, all their sizes are at most
$$
\exp\bigl(O_s(\log^{O(1)}(2K))\bigr),
$$
and all resulting approximate groups lie in $A^{O_s(1)}$ and have approximation parameter $K^{O_s(1)}$. Their generated subgroups are <abelian groups>.
There are $O(\log^{O(1)}(2K))$ factors at each of at most $s$ induction levels. Absorbing the resulting products of the bounds into the $O_s$ notation gives
$$
\boxed{m,n\leq O_s(\log^{O_s(1)}(2K)).}
$$
Keeping the factors in the order supplied by the induction yields the required product of the $X_i$ and $C_j$ containing $A$.
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