
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 uniform paint roller is used to shade the region around both diagonals of a square. The shaded area is two-thirds of the total area of the square. Determine the ratio of the side length of the square to the diameter of the roller.
A) \( \frac{4}{3} \)
B) \( \frac{3\sqrt{2}}{2} \)
C) \( 2\sqrt{2} \)
D) \( \frac{5\sqrt{2}}{2} \)

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