Find the value of \[ 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 + \cdots - 2020, \] where the sign alternates between $+$ and $-$ after every three numbers.

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