
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

A circle of radius \(2 \sqrt{5}\) is drawn through vertices \(A\) and \(B\) of triangle \(ABC\). This circle intersects line \(BC\) at a segment equal to \(4 \sqrt{5}\) and is tangent to line \(AC\) at point \(A\). From point \(B\), a perpendicular is drawn to line \(BC\) intersecting line \(AC\) at point \(F\). Find the area of triangle \(ABC\) if \(BF = 2\).

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