
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 that the incircle of triangle \( \triangle ABC \) touches sides \( AB \) and \( AC \) at points \( E \) and \( F \) respectively, and \( AD \) is the altitude from \( A \) to side \( BC \) of \( \triangle ABC \), and \( AE + AF = AD \). Determine the range of values for \( \sin \frac{A}{2} \).

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