
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
Color a set of $n$ equally spaced points on a circle either red or blue. Starting from any point and moving counterclockwise, record the colors of $k$ points $(k \leq n)$. This sequence of $k$ colors is called a "$k$-order color sequence". Two $k$-order color sequences are considered different if and only if they differ in at least one corresponding position. If any two "3-order color sequences" on a circle are different, what is the maximum number of equally spaced points $n$ on the circle?

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