
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, touching each other. The heights of the cones are the same, and the radii of their bases are \(2r\), \(3r\), and \(10r\). A ball with a radius of 2 is placed on the table, touching all three cones. It turns out that the center of the ball is equidistant from all points of contact with the cones. Find \(r\).

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