
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
From a quarter circle with a radius of $12 \text{ cm}$, a semicircle is cut out above one of the bounding radii, using it as its diameter. What is the radius of the largest circle that can be inscribed in the remaining shape?

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