
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
On a table, there are three cones standing on their bases and touching each other. The heights of the cones are equal, and the radii of their bases are $2r$, $3r$, and $10r$. A sphere of radius 2 is placed on the table, touching all the cones. It turns out that the center of the sphere is equidistant from all the points of contact with the cones. Find $r$.

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