
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
In square $PQRS$, points $L$ and $O$ lie on $\overline{PQ}$ and $\overline{PS}$ respectively, such that $PL = PO = x$. Points $M$ and $N$ lie on $\overline{QR}$ and $\overline{RS}$ respectively, with points $T$ and $U$ on $\overline{LO}$ such that $\overline{MT} \perp \overline{LO}$ and $\overline{NU} \perp \overline{LO}$. Assume triangle $PLQ$ and quadrilateral $QMTL$ each have area $1$, while quadrilateral $SNUL$ and pentagon $RNMUT$ each have area $2$. If the side of the square $PQRS$ is $3$, find $MT^2$.

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