
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 cones with a common apex $A$ touch each other externally, and the vertex angle of the first two cones is $\frac{\pi}{3}$. All cones also touch a plane passing through point $A$ and lie on the same side of it. Find the vertex angle of the third cone. (The vertex angle of a cone is the angle between its generators in the axial section.)

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