
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
Two spheres with a radius of 12 are placed on a table, touching each other externally. A cone touches the table with its lateral surface and also touches both spheres (externally). The distances from the apex of the cone to the points where the spheres touch the table are 13. Find the 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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