Are WO-type Wolf-Rayet (WR) stars in the final stage of massive star evolution before core-collapse? Although WC-and WO-type WRs have very similar spectra, WOs show a much stronger O vi λλ3811,34 emission-line feature. This has usually been interpreted to mean that WOs are more oxygen rich than WCs, and thus further evolved. However, previous studies have failed to model this line, leaving the relative abundances uncertain, and the relationship between the two types unresolved. To answer this fundamental question, we modeled six WCs and two WOs in the LMC using UV, optical, and NIR spectra with the radiative transfer code cmfgen in order to determine their physical properties. We find that WOs are not richer in oxygen; rather, the O vi feature is insensitive to the abundance. However, the WOs have a significantly higher carbon and lower helium content than the WCs, and hence are further evolved. A comparison of our results with single-star Geneva and binary BPASS evolutionary models show that, while many properties match, there is more carbon and less oxygen in the WOs than either set of evolutionary model predicts. This discrepancy may be due to the large uncertainty in the 12C+4He → 16O nuclear reaction rate; we show that if the Kunz et al. rate is decreased by a factor of 25%-50%, then there would be a good match with the observations. It would also help explain the LIGO/VIRGO detection of black holes whose masses are in the theoretical upper mass gap.