
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 vertex \( A \) are externally tangent to each other. The first two cones are identical, and the vertex angle of the third cone is \( \frac{\pi}{4} \). Each of these three cones is internally tangent to a fourth cone with vertex at point \( A \) and a vertex angle of \( \frac{3 \pi}{4} \). Find the vertex angle of the first two cones. (The vertex angle of a cone is the angle between its generatrices in its axial section.)

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