
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
A rectangular sheet of paper has an area of $12 \text{ cm}^2$ and its length is twice its width. The front is white and the back is black. When the sheet is folded so that point $A$ rests on the longer side as shown (not provided), the visible black area is equal to the visible white area. How many centimeters is $A$ from its original position? Express your answer in simplest radical form.

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