
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 sequence $\{a_n\}$ ($n=1,2,3,\ldots,2012$), and two circles $C_1: x^2+y^2-4x-4y=0$, $C_2: x^2+y^2-2a_nx-2a_{2013-n}y=0$. If circle $C_2$ bisects the circumference of circle $C_1$, then the sum of all terms of $\{a_n\}$ is \_\_\_\_\_\_.

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