
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
A convex $n$-gon ($n \geqslant 4$) is drawn on a board. Each of its vertices must be colored either black or white. We call a diagonal multicolored if its endpoints are colored differently. A vertex coloring is called good if the $n$-gon can be divided into triangles by multicolored diagonals that do not have common points (except for the vertices). Find the number of good colorings.

(S. Berlov)

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