
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
Triangle $ABC$ is inscribed in a circle $\omega$ with $AB = 6$, $BC = 8$, and $AC = 5$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. The circle $\gamma$ with diameter $\overline{DE}$ is tangent to circle $\omega$ at $E$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Therefore, find the exact value of $AF$.

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