
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 on the sides and the top. It is cut vertically into four pieces, such that one of the cut starts at the midpoint of the top edge and ends at a corner on the opposite edge. The piece whose top is triangular contains an area $A$ and is labeled as triangle $C$. Calculate the volume $v$ and the total surface area $a$ of icing covering this triangular piece. Compute $v+a$.
A) $\frac{24}{5}$ 
B) $\frac{32}{5}$ 
C) $8+\sqrt{5}$
D) $5+\frac{16\sqrt{5}}{5}$
E) $22.5$

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