
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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