We apply two non-perturbative methods, the numerical conformal bootstrap and supersymmetric localization, to four point functions of half-BPS operators in 3d maximally supersymmetric ABJM theory. This correlator is dual to scattering of gravitons and KK-modes in M-theory on AdS_4 x S^7, and determines the M-theory S-matrix in the flat space limit. Using localization, we compute OPE coefficients of certain protected operators exactly at small N and to all orders in 1/N at large N. We apply these analytic results to the numerical bootstrap in two ways. First, we find that numerical bootstrap bounds for these OPE coefficients are saturated by the analytic results, which allows us to read off all low-lying CFT data in the correlator, including for unprotected operators. Second, by imposing the analytical results we find precision islands in the space of certain quarter and eighth BPS OPE coefficients. This numerical data can be used to determine the M-theory S-matrix, which we confirm at leading order in large N.