My abstract is: Two-dimensional CFT correlators develop `forbidden singularities’ at large central charge c. These singularities signify information loss from black hole physics in AdS, arising as a consequence of Euclidean-time periodicity from the effective Hawking temperature. We study an infinite class of examples where these singularities can be resolved by non-perturbative effects at finite c, and we show that the resolution has certain universal features that apply to the general case. By Borel resumming the 1/c expansion of the exact examples, we explicitly identify `information-restoring’ effects from heavy states, which must correspond to classical solutions in AdS_3. In other words, exact correlators decompose into a piece corresponding to the perturbative AdS vacuum plus non-perturbative corrections from the exchange of trans-Planckian states, directly connecting unitarity to a sum over geometries. Our results also suggest a line of inquiry towards a more precise formulation of the gravitational path integral in AdS_3.