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Voyager 1 (V1), before it entered the local interstellar medium (LISM) in the summer of 2012, observed a series of increases and decreases in the galactic cosmic ray (GCR) nuclei (mainly protons with energies exceeding 70 MeV) fluxes. This remarkable feature was accompanied by the corresponding decreases and increases in the anomalous cosmic ray fluxes. We argue that these observations may be due to MHD instabilities and/or magnetic reconnection near the nose of the heliopause (HP), which created a possibility for V1 to cross the regions occupied by the solar wind and LISM plasma consecutively. It has been shown previously that the V1 observation of a jump-like increase in the GCR flux during its final penetration into the LISM could only occur if both pitch angle scattering and perpendicular diffusion were greatly reduced in the LISM. We demonstrate the effect of the solar wind and LISM turbulence, the properties of which are derived from 48 s averaged V1 measurements, on the HP instability. Since V1 was crossing the regions magnetically connected either to the Sun or to the LISM, the observed GCR fluxes were substantially affected. We present numerical simulations illustrating such behavior.