
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
Given a regular 12-sided polygon, draw non-intersecting diagonals to divide it into triangles in all possible ways such that each vertex of the polygon is a vertex of an odd number of the resulting triangles. (Two divisions are not considered different if one can be transformed into the other by rotation around the center or reflection across a line passing through the center.)

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