
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
Let $ABCD$, $AEFD$, and $ABGF$ be three adjacent square faces of a cube, where $AB = 12$, and $H$ is the eighth vertex of the cube. Let $I$, $J$, and $K$ be the points on $\overline{AB}$, $\overline{AF}$, and $\overline{AD}$, respectively, such that $AI = AF = AD = 3$. A solid $T$ is obtained by drilling a tunnel through the cube, where the sides of the tunnel are planes parallel to $\overline{DH}$, and containing edges $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $T$, including the walls of the tunnel, is expressed as $x + y\sqrt {z}$, where $x$, $y$, and $z$ are integers and $z$ is not divisible by the square of any prime. Determine $x + y + z$.

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