
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 \( \theta \) in the range \( (0, \pi) \), how many (incongruent) triangles \( ABC \) have angle \( A = \theta \), \( BC = 1 \), and have the following four points concyclic: \( A \), the centroid, the midpoint of \( AB \), and the midpoint of \( AC \)?

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