
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
Three spheres with radii 2, 2, and 1 are placed on a table, touching each other externally. The vertex \( C \) of a cone is located on the table, and the cone touches all three spheres externally. Point \( C \) is equidistant from the centers of the two equal spheres, while the cone touches the third sphere along a generatrix perpendicular to the table. Find the angle at the vertex of the cone. (The angle at the vertex of the cone is defined as the angle between its generatrices in an axial section.)

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