
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 $AB = 10$, and let $G$ be the eighth vertex opposite $A$ not lying on any of these faces. Points $P$, $Q$, and $R$, are on edges $\overline{AB}$, $\overline{AD}$, and $\overline{AE}$, respectively, such that $AP = AQ = AR = 3$. A solid $T$ is obtained by drilling a tunnel through the cube in the shape of a triangular prism with the sides of the tunnel being planes parallel to $\overline{AG}$ and containing $\overline{PQ}$, $\overline{QR}$, and $\overline{RP}$. Calculate the total surface area of $T$.

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