
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, the heights of which are half the length of their slant height. Three spheres are placed in the region, touching each other externally. Two of the spheres are identical and touch both cones, while the third one touches the boundary of the region. What is the maximum ratio of the radius of the third sphere to the radius of the first sphere?

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