
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 quadrilateral pyramid has a cube inscribed in it such that four of the cube's vertices are on the slant heights of the pyramid and four vertices lie in the plane of the base. All edges of the pyramid are equal, each having a length of $a$. Calculate the total surface area and the volume of the cube.

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