
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]


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A regular triangular prism is inscribed in a regular tetrahedron such that the vertices of one base of the prism lie on the lateral edges of the tetrahedron, and the vertices of the other base lie in the plane of the tetrahedron's base. The edge length of the tetrahedron is \(a\). Determine the volume of the prism if all its edges are equal.

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