
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
The region is bounded by two cones with a common base, whose height is 4 and base radius is 3. Three spheres are placed in the region, touching each other externally. Two of the spheres are identical and touch both cones, and the third touches the boundary of the region. What is the minimum radius of the third sphere?

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