
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 triangle \(ABC\), it is known that \(AC = 12\) and \(AB = BC = 3\sqrt{10}\). Two spheres touch the plane of triangle \(ABC\) at points \(A\) and \(C\), respectively, and are located on opposite sides of this plane. The distance between the centers of these spheres is 15. The center of a third sphere is at point \(B\), and this sphere touches the two given spheres externally. Find the radius of the third sphere.

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