
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
Consider a regular octagon centered at the origin in the complex plane with adjacent sides two units apart. One pair of sides is parallel to the real axis. Let \(R\) be the region outside the octagon, and let \(S = \left\{ \frac{1}{z} \mid z \in R \right\}\). Determine the area enclosed by \(S\).

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