
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 cubical cake with edge length $3$ inches is iced entirely (sides and the top). It is cut from a corner to the midpoint of the opposite face, slicing through three edges intersecting at the midpoint of the face diagonally opposite the selected corner. Calculate the sum of the cubical space and the area of the icing of the pyramidal segment thus created.

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