Wolf-Rayet stars of the carbon sequence (WC) represent the final evolutionary phase of very massive stars. They are the presumed immediate progenitors of black holes, stripped supernovae (Ibc SNe), and long duration gamma-ray bursts. With 28 WC stars, the Large Magellanic Cloud (LMC) is the only galaxy that hosts a resolvable and statistically significant population of WC stars at subsolar metallicity. Despite this, the physical parameters of most remain unknown, and, in stark contrast to prediction: only 3/28 of the LMC WC stars have been confirmed as binaries. However, previous studies were severely insensitive to year-long orbital periods that WC binaries are expected to exhibit. We propose to obtain single-epoch high-resolution COS/FUV spectra to employ a novel method for binary detection and to derive the wind parameters of the WC population and their potential companions. This will serve as an indispensable step for establishing the highly uncertain onward evolution of very massive stars in the era of gravitational-wave astronomy.