
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
Let complex number \( z \) be four-presentable if there exists a complex number \( w \) of absolute value \( 4 \) such that \( z = w - \frac{1}{w} \). Define \( S \) as the set of all four-presentable complex numbers. Determine the area inside the closed curve formed by \( S \) in the complex plane.

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