
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
Given the triangle \(\triangle ABC\) is inscribed in the unit circle, and the internal angle bisectors of angles \(A\), \(B\), and \(C\) intersect the circle again at points \(A_{1}\), \(B_{1}\), and \(C_{1}\) respectively, evaluate the expression 
$$
\frac{A A_{1} \cdot \cos \frac{A}{2} + B B_{1} \cdot \cos \frac{B}{2} + C C_{1} \cdot \cos \frac{C}{2}}{\sin A + \sin B + \sin C}$$

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