
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
In triangle \(ABC\), a perpendicular bisector to the side \(AB\) is drawn to intersect another side at some point \(C'\). Similarly, points \(A'\) and \(B'\) are constructed. For which original triangles will the triangle \(A'B'C'\) be equilateral?

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