
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 $ABCD$, $ABFE$, and $ADHE$ be three adjacent square faces of a cube, where $AE = 10$, and let $G$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{AB}$, $\overline{AD}$, and $\overline{AE}$, respectively, so that $AI = AJ = AK = 3$. A solid $T$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AG}$, and containing the edges, $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $T$, including the walls of the tunnel, is $s + t\sqrt {u}$, where $s$, $t$, and $u$ are positive integers and $u$ is not divisible by the square of any prime. Find $s + t + u$.

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