
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 cone with a base radius of 2 and a slant height of \( c \) contains three spheres, each with radius \( r \). The spheres touch each other (externally), the lateral surface of the cone, and the first two spheres touch the base of the cone. Find the maximum value of \( r \).

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