
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
**Square $PQRS$ has sides of length 2. Point $T$ is on $\overline{QR}$ such that $QT = \frac{2}{3}QR$, and Point $U$ is on $\overline{RS}$ such that $\triangle PTU$ is a right triangle at $T$. A square with one side on $\overline{PU}$ and another vertex on line $\overline{PS}$ has vertices $P'$, $Q'$, $R'$, and $S'$, with $S'$ on $\overline{PS}$. Determine the length of a side of this smaller square.**

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