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The blazar Mrk 501 is among the brightest X-ray and TeV sources in the sky, and among the few sources whose spectral energy distributions can be characterized by current instruments with relatively short observations (minutes to hours). In 2013, we organized an extensive multi-instrument campaign including Fermi-LAT, MAGIC, VERITAS, F-GAMMA, Swift, GASP-WEBT, and other groups and instruments which provided the most detailed temporal and energy coverage on Mrk 501 to date. This campaign included, for the first time, observations with the Nuclear Stereoscopic Telescope Array (NuSTAR), a satellite mission launched in 2012. NuSTAR provides unprecedented sensitivity in the hard X-ray range 3-79 keV, which, together with very high energy (VHE; >100 GeV) observations, is crucial to probe the highest energy electrons in Mrk 501. The campaign covered a few day long flaring activity in July 2013 which could be studied with strictly simultaneous NuSTAR and MAGIC observations. A large fraction of the MAGIC data during this activity was affected by hazy atmospheric conditions, due to a sand layer from the Saharan desert. These data would have been removed in any standard Cherenkov Telescope data analysis. MAGIC has developed a technique to correct for adverse atmospheric conditions, making use of information from the LIDAR facility at the MAGIC site, and applies an event-by-event correction to recover data affected by adverse weather conditions. This is the first time that LIDAR information has been used to produce a physics result with Cherenkov Telescope data taken during adverse atmospheric conditions, and hence sets a precedent for current and future ground-based gamma-ray instruments. In this contribution we report the observational results, focusing on the LIDAR-corrected MAGIC data and the strictly simultaneous NuSTAR and MAGIC/VERITAS data, and discuss the scientific implications.

First study of Mrk501 through the eyes of NuSTAR, VERITAS and the $\textit{LIDAR-corrected}$ eyesight of MAGIC