
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
Right triangle $PQR$ has a leg $PQ = 9$ cm, another leg $PR = 12$ cm, and a right angle at $P$. A square is fitted such that one of its sides lies along the hypotenuse $QR$, and each of its other vertices touches one of the two legs of the triangle. Determine the length of a side of the square in centimeters, expressed as a common fraction.

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