
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 a right triangle $PQR$, where angle $P$ is the right angle, the lengths of legs are 9 cm (for leg $PR$) and 12 cm (for leg $PQ$). A square has one side along the hypotenuse of triangle $PQR$ and one vertex each on the legs $PR$ and $PQ$. Determine the side length of the square.

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