
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 $ABC$, inscribed in circle $\omega$, the sides are $AB=6$, $BC=8$, and $AC=4$. The angle bisector of $\angle A$ intersects side $BC$ at $D$ and circle $\omega$ again at point $E$. A circle $\gamma$, with diameter $DE$, intersects $\omega$ a second time at point $F$. Calculate $AF^2$ and express it as a fraction $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find the sum $p+q$.

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