
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 $AD = 10$, and let $G$ be the eighth vertex of the cube. Let $L$, $M$, and $N$, be the points on $\overline{AB}$, $\overline{AD}$, and $\overline{AG}$, respectively, so that $AL = AM = AN = 3$. A solid $T$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{BG}$, and containing the edges, $\overline{LM}$, $\overline{MN}$, and $\overline{NL}$. The surface area of $T$, including the walls of the tunnel, is $q + r\sqrt{s}$, where $q$, $r$, and $s$ are positive integers and $s$ is not divisible by the square of any prime. Find $q + r + s$.

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