Compute the sum \[ S = 1 + 2 + 3 - 4 - 5 + 6 + 7 + 8 - 9 - 10 + \dots - 2010 \] where every three consecutive additions (\(+\)) are followed by two subtractions (\(-\)).

Let's think step by step and output the final answer within \boxed{}.
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