
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 $\omega$, side lengths are given as $AB=6$, $BC=8$, and $AC=4$. The bisector of angle $A$ intersects side $\overline{BC}$ at $D$ and circle $\omega$ again at $E$. Define circle $\gamma$ with diameter $\overline{DE}$. Circles $\omega$ and $\gamma$ intersect at $E$ and another point $F$. Calculate $AF^2$ and express it as a fraction $\frac{m}{n}$, where $m$ and $n$ are coprime integers. Find the sum $m+n$.

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