
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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Triangle \( \triangle ABC \) is inscribed in the unit circle. The internal angle bisectors of angles \( A \), \( B \), and \( C \) are extended and intersect the circle at points \( A_1 \), \( B_1 \), and \( C_1 \) respectively. Determine the value of \( \frac{A A_1 \cos \frac{A}{2} + B B_1 \cos \frac{B}{2} + C C_1 \cos \frac{C}{2}}{\sin A + \sin B + \sin C} \).

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