
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 \(ABC\), \(\angle BAC = 60^\circ\). Let \(\omega\) be a circle tangent to segment \(AB\) at point \(D\) and segment \(AC\) at point \(E\). Suppose \(\omega\) intersects segment \(BC\) at points \(F\) and \(G\) such that \(F\) lies between \(B\) and \(G\). Given that \(AD = FG = 4\) and \(BF = \frac{1}{2}\), find the length of \(CG\).

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