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In this paper, we propose a continuation method for box constrained variational inequality problems. The continuation method is based on the class of Gabriel-Moré smooth functions and has the following attractive features: It can start from any point; It has a simple and natural neighborhood definition; It solves only one approximate Newton equation at each iteration; It converges globally linearly and locally quadratically under nondegeneracy assumption at the solution point and other suitable assumptions. A hybrid method is also presented, which is shown to preserve the above convergence properties without the nondegeneracy assumption at the solution point. In particular, the hybrid method converges finitely for affine problems.