
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 vertices of a regular  $2012$ -gon are labeled  $A_1,A_2,\ldots, A_{2012}$  in some order. It is known that if  $k+\ell$  and  $m+n$  leave the same remainder when divided by  $2012$ , then the chords  $A_kA_{\ell}$  and  $A_mA_n$  have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled  $A_1$  and  $A_4$ . How is the tenth vertex labeled?

*Proposed by A. Golovanov*

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