
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
In a cone with a base radius of 2 and a slant height of \( c \), three spheres with radius \( r \) are placed. These spheres touch each other externally, touch 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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