
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 $M$ and $T$ lie on $\overline{PQ}$ and $\overline{PS}$, respectively, such that $PM=PT$. Points $N$ and $U$ lie on $\overline{QR}$ and $\overline{RS}$, respectively, and points $V$ and $W$ lie on $\overline{TM}$ such that $\overline{NV} \perp \overline{TM}$ and $\overline{UW} \perp \overline{TM}$. Triangle $PMT$, quadrilateral $QNVM$, quadrilateral $STWU$, and pentagon $RUVWN$ each has area $1$. Determine $NV^2$.

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