
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
A sphere is inscribed in a right circular cone with base radius \(10\) cm and height \(20\) cm. Find the radius \(r\) of the sphere expressed in the form \(a \sqrt{c} - a\). What is \(a + c\)?

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