
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
Consider a cube with vertex labels $EFGH$, $EFDC$, and $EHBC$, where $EC = 12$. Let $A$ be the opposite vertex of the cube not connected directly to $E$, $F$, $G$, or $H$. Define points $I$, $J$, and $K$ on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$ respectively, such that $EI = EJ = EK = 3$. A solid $S'$ is then defined by removing a tunnel through the cube where the sides of the tunnel are planes parallel to $\overline{AE}$, incorporating the edges $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. Compute the surface area of $S'$, including the walls of the tunnel, and express it in the form $m + n\sqrt p$ where $m$, $n$, and $p$ are integers with $p$ square-free.

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