
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
The base of the triangular pyramid \( SABC \) is an equilateral triangle \( ABC \) with a side length of 8. The lateral edge \( SC \) is perpendicular to the base and has a length of 15. A sphere, whose center \( O \) lies in the plane \( SBC \), touches the edges \( SA \), \( AB \), and \( AC \) at points \( A_1 \), \( B_1 \), and \( C_1 \) respectively. Find \( AA_1 \), the distance from point \( O \) to edge \( BC \), and the radius of the sphere.

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