
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 regular hexagon with center at the origin in the complex plane has opposite pairs of sides two units apart. One pair of sides is parallel to the imaginary axis. Let \( R \) be the region outside the hexagon, and let \( S = \{ \frac{1}{z} \mid z \in R \} \). Find the area of \( S \).

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