
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
The radius of a sphere inscribed in a regular triangular pyramid is four times smaller than the side of the pyramid's base. Find the cosine of the dihedral angle at the pyramid's apex.

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