
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 cube with an edge length of 3 inches is iced on all its sides and the top. The cube is cut vertically into three pieces, with a cut passing from the midpoint of one of the top edges to an opposite corner, dividing the top face into different shapes. One of these pieces, whose top view is a triangular region ABC, where A is a corner of the cube, B is the midpoint of the edge opposite to A, and C is the adjacent corner. Determine the sum of the volume c cubic inches of this triangular piece and the area s square inches of icing covering it.

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