We analyze the performance of Belief Propagation Guided Decimation, a physics-inspired message passing algorithm, on the random $k$-XORSAT problem. Specifically, we derive an explicit threshold up to which the algorithm succeeds with a strictly positive probability Ω(1) that we compute explicitly, but beyond which the algorithm with high probability fails to find a satisfying assignment. In addition, we analyze a thought experiment called the decimation process for which we identify a (non-)reconstruction and a condensation phase transition. The main results of the present work confirm physics predictions from [Ricci-Tersenghi and Semerjian: J. Stat. Mech. 2009] that link the phase transitions of the decimation process with the performance of the algorithm, and improve over partial results from a recent article [Yung: Proc. ICALP 2024].