
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 $LMNOP$, points $Q$ and $T$ lie on $\overline{LM}$ and $\overline{PL}$, respectively, so that $LQ=LT.$ Points $R$ and $S$ lie on $\overline{MN}$ and $\overline{OP}$, respectively, and points $U$ and $V$ lie on $\overline{QT}$ so that $\overline{RU} \perp \overline{QT}$ and $\overline{SV} \perp \overline{QT}$. The areas of triangle $LQT$, quadrilateral $MRUQ$, quadrilateral $PSVT$, and pentagon $NRSUV$ each equal $1.$ Determine $RU^2$.

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