A$ of size at most $\exp(\log^{O(1)}2\tilde K)$ such that $$\label{eq:induction.step}
\tilde A\subset X\tilde A_1\cdots\tilde A_r.$$ Since $G/N_0$ is generated by the $K$-approximate group $\rho(A)$, we may apply the induction hypothesis to each approximate subgroup $\rho(\tilde A_i)$ of $G/N_0$ to obtain, for each $i=1,\ldots,r_0$, integers $$\begin{aligned}
r_i,\ell_i&\le e^{O((\tilde s-1)^2)}\log^{O(\tilde s-1)}(2\tilde K^{e^{O(1)}})\\
   & \le e^{O(\tilde s(\tilde s-1))}\log^{O(\tilde s-1)}2\tilde K;\end{aligned}$$ a normal subgroup $N_i\lhd G$ containing $N_0$ and satisfying $$\begin{aligned}
N_i&\subset A^{e^{O((\tilde s-1)^2)}K^{e^{O(s)+O(\tilde s-1)}(e^{O(1)}m)}(e^{O(1)}\log2\tilde K)^{O(\tilde s-1)}}N_0\\
  &\subset A^{e^{O(\tilde s(\tilde s-1))}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s-1)}2\tilde K}N_0;\end{aligned}$$ finite $\tilde K^{e^{O(\tilde s)}}$-approximate groups $A_1^{(i)},\ldots,A_{r_i}^{(i)}\subset\tilde A_i^{e^{O(\tilde s-1)}}\subset\tilde A^{e^{O(\tilde s)}}$ such that, writing $\pi_i:G\to G/N_i$ for the quotient homomorphism, each group $\langle\pi_i(A_j^{(i)})\rangle$ is abelian; and sets $X_1^{(i)},\ldots,X_{\ell_i}^{(i)}\subset\tilde A_i^{e^{O(\tilde s-1)}}\subset\tilde A^{e^{O(\tilde s)}}$ satisfying $$\begin{aligned}
|X_j^{(i)}|&\le\exp(e^{O(\tilde s-1)}\log^{O(1)}(2\tilde K^{e^{O(1)}}))\\
   &\le\exp(e^{O(\tilde s)}\log^{O(1)}2\tilde K)\end{aligned}$$ such that $$\label{eq:induction.hyp}
\tilde A_i\subset N_i\prod\{A_1^{(i)},\ldots,A_{r_i}^{(i)},X_1^{(i)},\ldots,X_{\ell_i}^{(i)}\},$$ with the product taken in some order.

Defining $N=N_1\cdots N_{r_0}$, we then have $$\begin{aligned}
N&\subset A^{r_0e^{O(\tilde s(\tilde s-1))}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s-1)}2\tilde K}\cdot N_0\\
   &\subset A^{e^{O(\tilde s(\tilde s-1))}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s)}2\tilde K}\cdot N_0\\
   &\subset A^{e^{O(\tilde s(\tilde s-1))}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s)}2\tilde K}\cdot A^{K^{e^{O(s)}m}}\\
   &\subset A^{e^{O(\tilde s^2)}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\ti