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Neptune-size exoplanets seem particularly sensitive to atmospheric evaporation, making it essential to characterize the stellar high-energy radiation that drives this mechanism. This is particularly important with M dwarfs, which emit a large and variable fraction of their luminosity in the ultraviolet and can display strong flaring behavior. The warm Neptune GJ 3470b, hosted by an M2 dwarf, was found to harbor a giant exosphere of neutral hydrogen thanks to three transits observed with the Hubble Space Telescope Imaging Spectrograph (HST/STIS). Here we report on three additional transit observations from the Panchromatic Comparative Exoplanet Treasury program, obtained with the HST Cosmic Origin Spectrograph. These data confirm the absorption signature from GJ 3470b’s exosphere in the stellar Lyman-α line and demonstrate its stability over time. No planetary signatures are detected in other stellar lines, setting a 3σ limit on GJ 3470b’s far-ultraviolet (FUV) radius at 1.3 times its Roche lobe radius. We detect three flares from GJ 3470. They show different spectral energy distributions but peak consistently in the Si III line, which traces intermediate-temperature layers in the transition region. These layers appear to play a particular role in GJ 3470’s activity as emission lines that form at lower or higher temperatures than Si III evolved differently over the long term. Based on the measured emission lines, we derive synthetic X-ray and extreme-ultraviolet (X+EUV, or XUV) spectra for the six observed quiescent phases, covering one year, as well as for the three flaring episodes. Our results suggest that most of GJ 3470’s quiescent high-energy emission comes from the EUV domain, with flares amplifying the FUV emission more strongly. The neutral hydrogen photoionization lifetimes and mass loss derived for GJ 3470b show little variation over the epochs, in agreement with the stability of the exosphere. Simulations informed by our XUV spectra are required to understand the atmospheric structure and evolution of GJ 3470b and the role played by evaporation in the formation of the hot-Neptune desert.