
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
Divide a circle into \( n \) equal parts and color each point either red or blue. Starting from any point, record the colors of \( k \) points in a counterclockwise direction, which is called a " \( k \)-order color sequence." Two \( k \)-order color sequences are considered different if there is at least one position where the colors differ. If any two 3-order color sequences are different, what is the maximum value of \( n \)?

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