
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 nonagon are colored such that: 
1) Adjacent vertices are different colors, and 
2) If 3 vertices form an equilateral triangle, they are all different colors.
Let \( m \) be the minimum number of colors needed for a valid coloring, and \( n \) be the total number of colorings using \( m \) colors. Determine \( mn \). (Assume each vertex is distinguishable.)

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