
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 \( A \) and generator length \( \sqrt{8} \) touch each other externally. For two cones, the angle between the generator and the axis of symmetry is \( \frac{\pi}{6} \), and for the third, it is \( \frac{\pi}{4} \). Find the volume of the pyramid \( O_{1} O_{2} O_{3} A \), where \(O_{1}, O_{2}, O_{3}\) are the centers of the bases of the cones.

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