= Solution
The intended quantity is $\boxed{N_{\rm th}^*}$, the threshold <population inversion> density of the <optical cavity>[cavity] after switching to low loss. At this value stimulated gain exactly balances the photon-loss rate:
$$
\left.\frac{dn}{dt}\right|_{N^*=N_{\rm th}^*}=0.
$$
Above threshold a nonzero seed grows; below threshold it decays. The unstarred $N_{\rm th}$ in the subpart is shorthand for the starred threshold quantity in the displayed equations, rather than a different total atomic population. The <photon lifetime> $\tau_c$ fixes the <photon number density>[photon-density] loss rate $n/\tau_c$.
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