
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 rectangular cake with dimensions $4$ inches, $3$ inches, and $2$ inches (length, width, height respectively) is iced on the sides, the top, and the bottom. The cake is cut from the top center vertex across to the midpoint of the bottom edge on the opposite side face, creating one triangular piece. If the top center vertex is $T$, and the midpoint on the opposite bottom edge is denoted as $M$, find the volume of the cake on one side of the cut ($c$ cubic inches) and the area of icing on this piece ($s$ square inches). Calculate the sum $c+s$.

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