
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
Let $\mathcal{C}$ be a circle in the \(xy\) plane with radius 1 and center \((0,0,0)\), and let \(P\) be a point in space with coordinates \((3,4,8)\). Find the largest possible radius of a sphere that is contained entirely in the slanted cone with base \(\mathcal{C}\) and vertex \(P\).

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