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The origin of cosmic rays above the knee in the spectrum is an unsolved problem. We present a wind model in which interstellar gas flows along a non-rotating, expanding flux tube with a changing speed and cross-sectional area. Cosmic rays from Galactic sources, such as supernova remnants, which are coupled to the plasma via Alfvén waves, provide the main pressure source for driving this outflow. These cosmic rays are then subject to diffusive shock reacceleration at the Galactic wind termination shock, which is located at a distance ~200 kpc. Some of the highest-energy reaccelerated particles propagate upstream against the wind and can contribute to the PeV-EeV range of the spectrum. We analyze the conditions under which efficient reacceleration can occur and find that rigidities ~10-40 PV can be obtained and that the termination shock may account for half of the proton spectrum measured in IceCube/IceTop experiment. The highest-energy particles that escape downstream from our termination shock, and similar shocks surrounding most galaxies, can be further accelerated by intergalactic shock fronts.