
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
There are two spheres on a table, touching each other externally. A cone touches the table with its lateral surface and also touches both spheres externally. The apex of the cone lies on the segment connecting the points where the spheres touch the table. It is known that the rays connecting the apex of the cone with the centers of the spheres form equal angles with the table. Find the maximum possible angle at the apex of the cone. (The angle at the apex of the cone is defined as the angle between its generatrices in the axial section.)

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