
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 regular triangle is inscribed in a circle. Find the set of intersection points of the altitudes of all possible triangles inscribed in the same circle, where two sides of each triangle are parallel to two sides of the given regular triangle.

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