
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 the function $f(x) = |x-2012| + \ldots + |x-2| + |x-1| + |x+1| + |x+2| + \ldots + |x+2012| \, (x \in \mathbb{R})$, and $f(a^2+2a+2) > f(a)$, then the range of real numbers $a$ that satisfy the condition is.

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