
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

Three cones are standing on their bases on a table, touching each other. The radii of their bases are 32, 48, and 48, and the apex angles are \(\frac{\pi}{3}\), \(\frac{2\pi}{3}\), and \(\frac{2\pi}{3}\) respectively (the apex angle of a cone is the angle between its generatrices in an axial section). A sphere is suspended above the table, touching all the cones. It turns out that the center of the sphere is equidistant from the centers of the bases of all the cones. Find the radius of the sphere.

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