
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
Given a complex number $z$ is four-presentable if there is a complex number $w$ of absolute value $4$ such that $z = w - \frac{1}{w}$. Let $U$ be the set of all four-presentable complex numbers. This set $U$ forms a potentially closed curve in the complex plane. What is the area inside $U$?

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