
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 identical cones with apex \( A \) and an apex angle of \(\frac{\pi}{3}\) touch each other externally. Each of these cones touches internally a fourth cone with apex at point \( A \). Find the apex angle of the fourth cone. (The apex angle of a cone is defined as the angle between its generatrices in an axial section.)

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