
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 $\triangle ABC$, let \( D, E, F \) be the midpoints of \( BC, AC, AB \) respectively, and let \( G \) be the centroid of the triangle. For each value of \(\angle BAC\), how many non-similar triangles are there in which \( AEGF \) is a cyclic quadrilateral?

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