
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 $D$, $E$, and $F$ be the midpoints of $BC$, $CA$, and $AB$, respectively. $G$ is the centroid of the triangle. How many non-similar triangles $ABC$ are there for each value of $\angle BAC$ so that $AEGF$ is a cyclic quadrilateral?

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