
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
Inside a right circular cone with base radius \(6\) and height \(15\) are three congruent spheres with radius \(r\). Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. Determine the value of \(r\).
A) \(\frac{45}{3 + 5\sqrt{3} + \frac{\sqrt{261}}{2}}\)
B) \(\frac{90 - 40\sqrt{3}}{11}\)
C) \(\frac{80 + 35\sqrt{3}}{18}\)
D) 4
E) \(\frac{5}{2}\)

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