
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 at point $A$ touch each other externally, with the first two cones being identical and the third cone having an apex angle of $2 \arcsin \frac{1}{4}$. Each of these cones touches a fourth cone internally, which also has its apex at point $A$. Find the apex angle of the first two cones, given that it is half the apex angle of the fourth cone. (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}
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