
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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?

*Proposed by Aaron Lin*

Present the answer in LaTex format: \boxed{Your answer}
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