
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 $PQRS$, $PQVT$, and $PRUT$ be three adjacent square faces of a cube, for which $PT = 10$, and let $O$ be the eighth vertex of the cube. Let $L$, $M$, and $N$, be the points on $\overline{PQ}$, $\overline{PR}$, and $\overline{PT}$, respectively, so that $PL = PM = PN = 3$. A solid $X$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{OP}$, and containing the edges, $\overline{LM}$, $\overline{MN}$, and $\overline{NL}$. The surface area of $X$, including the walls of the tunnel, is $u + v\sqrt {w}$, where $u$, $v$, and $w$ are positive integers and $w$ is not divisible by the square of any prime. Find $u + v + w$.

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