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Contrary to expectations from current stellar evolutionary models recentobservations uncovered the ejection of shells of material by massive stars inthe years before the supernova (SN) explosion. The physical mechanism behind theimpulsive mass ejection synchronized with the stellar core-collapse is unclear.Here we propose a coordinated XMM-NuSTAR effort to map the evolution of thebroad-band X-ray spectrum of 1 nearby (d and amp lt 50 Mpc) or X-ray bright (Fx and amp gt 3 x 10^-13erg/s/cm2) strongly interacting SN and enable progress. Our program has theimmediate goal to characterize the medium around strongly interacting SNe whichoriginate from stellar progenitors with the most extreme mass loss beforeexplosion. [truncated! Please see actual data for full text]