
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
Triangle $PQR$ is a right triangle with the right angle at $P$. It has one leg $PQ$ of length 9 cm, and another leg $PR$ of length 12 cm. A square is constructed such that its one side lies on the hypotenuse $QR$ of triangle $PQR$, and one vertex on each of the legs $PQ$ and $PR$. Determine the length of one side of the square, expressed as a common fraction.

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