
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 equilateral triangle \(ABC\), a circle \(\omega\) is drawn such that it is tangent to all three sides of the triangle. A line is drawn from \(A\) to point \(D\) on segment \(BC\) such that \(AD\) intersects \(\omega\) at points \(E\) and \(F\). If \(EF = 4\) and \(AB = 8\), determine \(|AE - FD|\).

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