
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 $P$, $Q$, and $R$, be the points on $\overline{AB}$, $\overline{AD}$, and $\overline{AG}$, respectively, so that $AP = AQ = AR = 3$. A solid $T$ is created by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{BG}$, containing the edges $\overline{PQ}$, $\overline{QR}$, and $\overline{RP}$. The surface area of $T$, including the walls of the tunnel, is expressed as $x + y\sqrt{z}$, where $x$, $y$, and $z$ are positive integers and $z$ is not divisible by the square of any prime. Find $x + y + z$.

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