
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 a table, there are two spheres with radii 4 and 1, with centers \(O_{1}\) and \(O_{2}\), respectively, touching each other externally. A cone touches the side surface of the table and both spheres (externally). The apex \(C\) of the cone is located on the segment connecting the points where the spheres touch the table. It is known that the rays \(CO_{1}\) and \(CO_{2}\) form equal angles with the table. 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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