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

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