
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 each lie on a table, touching each other externally. A cone touches the table with its lateral surface and both spheres externally. The distances from the vertex of the cone to the points where the spheres touch the table are both 13. Find the angle at the apex of the cone. (The angle at the apex of the cone is the angle between its generatrices in an axial cross-section.)

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