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

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