
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