
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
On the faces of a regular tetrahedron with edge length $a$, equal regular pyramids are constructed, each with their bases on the faces of the tetrahedron. The dihedral angles at the vertices of these pyramids, opposite the faces of the tetrahedron, are right angles. Consider the polyhedron formed by the tetrahedron and the constructed pyramids. How many faces does this polyhedron have? What is it called?

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