
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 right triangle $ABC$, where $A$ is the right angle, leg $AB$ is 9 cm, and leg $AC$ is 12 cm. A square is inscribed such that one of its sides lies along the hypotenuse $BC$ and each of the remaining vertices touches one of the other two sides of the triangle. Determine the length of a side of the square, in cm, expressed as a common fraction.

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