
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 regular tetrahedron with edge length $a$ is given. Find the volume of the polyhedron formed by the intersection of this tetrahedron with its image under symmetry relative to the midpoint of its height.

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