
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 circle $\omega$ with $AB=6$, $BC=8$, and $AC=4$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. Let $M$ be the midpoint of $AB$. The circle passing through $B, D, M$ intersects $\omega$ again at a point $F$ distinct from $B$. Then calculate $AF^2.$

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