
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]


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In triangle \( ABC \) with the usual notations, \( a > b \). Additionally, for the internal and external angle bisectors at vertex \( C \), the lengths of the segments extending to the line \( AB \) are \( f_{1} \) and \( f_{2} \) respectively. These lengths satisfy the equation:

$$
\frac{f_{2}}{f_{1}} = \frac{a+b}{a-b} \sqrt{3}
$$

Determine the angle at vertex \( C \).

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