
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 vertex \(A\) touch each other externally, where the first two are identical, and the third one has an angle at the vertex equal to \(\frac{\pi}{2}\). All the cones also touch a plane passing through point \(A\) and lie on one side of it. Find the angle at the vertex for the first two cones. (The angle at the vertex of a cone is defined as the angle between its generators in the axial section.)

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