
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
Karabas-Barabas has a large plot of land in the shape of a convex $12$-gon, with lanterns placed at each vertex. Karabas-Barabas needs to place a finite number of additional lanterns inside the plot, divide the plot into triangular sections with vertices at the lanterns, and distribute these sections to the actors of the theater. Each internal lantern must illuminate at least six triangular sections (each lantern illuminates only the sections for which it is a vertex). What is the maximum number of triangular sections that Karabas-Barabas can distribute to the actors?

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