
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
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. If $(a_2-1)^3 + 2012\cdot(a_2-1) = 1$, and $(a_{2011}-1)^3 + 2012\cdot(a_{2011}-1) = -1$, then among the following four statements, the correct ones are \_\_\_\_\_\_\_\_.
1. $S_{2011} = 2011$
2. $S_{2012} = 2012$
3. $a_{2011} < a_2$
4. $S_{2011} < S_2$

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