
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
On the table lie two spheres with a radius of 4, touching each other externally. A cone touches the table along its lateral surface and both spheres externally. The distances from the apex of the cone to the points where the spheres touch the table are 5. Find the vertex angle of the cone. (The vertex angle of the cone is the angle between its generatrices in an axial section.)

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