
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
Let  $n \geqslant 3$  be integer. Given convex  $n-$ polygon  $\mathcal{P}$ . A  $3-$ coloring of the vertices of  $\mathcal{P}$  is called *nice* such that every interior point of  $\mathcal{P}$  is inside or on the bound of a triangle formed by polygon vertices with pairwise distinct colors. Determine the number of different nice colorings. 
(*Two colorings are different as long as they differ at some vertices.*)

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