The expression $\circ 1 \circ 2 \circ 3 \circ \dots \circ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?

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