
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 $ABC$ has one leg of length 9 cm, another leg of length 12 cm, and a right angle at $A$. A rectangle is placed such that one side lies on the hypotenuse of triangle $ABC$, and a vertex on each of the two legs of triangle $ABC$. The long side of the rectangle is twice the length of the short side. What is the length of the short side of the rectangle?

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