Let us denote
$$\sigma_\ell:=\sum_{i=1}^\ell q_i\qquad\text{and}\qquad\tau_\ell:=\sum_{i=\ell+1}^s\frac{1}{q_i}.$$
Then
$$\prod_{\ell=1}^{s-1}\left(\frac{q_\ell}{q_{\ell+1}}\cdot\frac{\sigma_{\ell+1}}{\sigma_\ell}\cdot\frac{\tau_{\ell-1}}{\tau_\ell}\right)=\frac{q_1}{q_s}\cdot\frac{\sigma_{s}}{\sigma_1}\cdot\frac{\tau_0}{\tau_{s-1}}=\sigma_s\tau_0>n,$$
hence there exists $\ell\in\{1,2,\dotsc,s-1\}$ such that
$$\frac{q_\ell}{q_{\ell+1}}\cdot\frac{\sigma_{\ell+1}}{\sigma_\ell}\cdot\frac{\tau_{\ell-1}}{\tau_\ell}>n^{1/s}=2^{1000}.$$
On the other hand, the left-hand side equals
\begin{align*}
\frac{q_\ell}{q_{\ell+1}}\cdot\frac{\sigma_{\ell+1}}{\sigma_\ell}\cdot\frac{\tau_{\ell-1}}{\tau_\ell}&=\frac{q_\ell}{q_{\ell+1}}\left(1+\frac{q_{\ell+1}}{\sigma_\ell}\right)\left(1+\frac{1}{q_\ell\tau_\ell}\right)\\
&=\frac{q_\ell}{q_{\ell+1}}+\frac{q_\ell}{\sigma_\ell}+\frac{1}{q_{\ell+1}\tau_\ell}+\frac{1}{\sigma_\ell\tau_\ell},
\end{align*}
so for the same $\ell$ we have
$$\frac{q_\ell}{q_{\ell+1}}+\frac{q_\ell}{\sigma_\ell}+\frac{1}{q_{\ell+1}\tau_\ell}+\frac{1}{\sigma_\ell\tau_\ell}>2^{1000}.$$
The first three terms on the left-hand do not exceed $1$, hence in fact
$$\sigma_\ell\tau_\ell\leq\frac{1}{2^{1000}-3}.$$