
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 circle $\omega$, triangle $ABC$ is inscribed with sides $AB = 6$, $BC = 8$, and $AC = 5$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Construct circle $\gamma$ with diameter $\overline{DF}$. Circles $\omega$ and $\gamma$ intersect at $F$ and a second point $E$. If triangle $AFE$ is isosceles with $AF=FE$, find the square of the length $AF^2$.

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