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In normal metals, macroscopic properties are understood using the concept of quasiparticles. In the cuprate high-temperature superconductors, the metallic state above the highest transition temperature is anomalous and is known as the "strange metal." We studied this state using angle- resolved photoemission spectroscopy. With increasing doping across a temperature-independent critical value p(c) similar to 0.19, we observed that near the Brillouin zone boundary, the strange metal, characterized by an incoherent spectral function, abruptly reconstructs into a more conventional metal with quasiparticles. Above the temperature of superconducting fluctuations, we found that the pseudogap also discontinuously collapses at the very same value of p(c). These observations suggest that the incoherent strange metal is a distinct state and a prerequisite for the pseudogap; such findings are incompatible with existing pseudogap quantum critical point scenarios.