
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
1.	Around a circumference are written  $2012$  number, each of with is equal to  $1$  or  $-1$ . If there are not  $10$  consecutive numbers that sum  $0$ , find all possible values of the sum of the  $2012$  numbers.

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