
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
Given a pyramid with a regular hexagonal base and lateral faces that are isosceles triangles with a vertex angle of $60^{\circ}$, find the volume of a regular tetrahedron placed within the pyramid such that one vertex is at the center of the hexagonal base and the other three vertices touch the midpoints of three non-adjacent edges of the hexagon.

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