
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 $A$ touch each other externally, with the first two being identical, and the third one having an apex angle of $\frac{\pi}{3}$. Each of the cones touches internally a fourth cone with a vertex at point $A$ and an apex angle of $\frac{5 \pi}{6}$. Find the apex angle $y$ of the first two cones. (The apex angle of a cone is defined as the angle between its generatrices in the axial section.)

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