
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
A right pyramid has an equilateral triangular base with a perimeter of 24 inches. Its apex is 8 inches vertically above each vertex of the base. Calculate the vertical height of the pyramid from its apex to the centroid of its triangular base.

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