
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 octagon is centered at the origin in the complex plane with opposite pairs of sides two units apart. One pair of sides is parallel to the real axis. Let $T$ be the region outside the octagon, and let $U = \left\lbrace\frac{1}{z} \ | \ z \in T\right\rbrace$. Find the area of $U.$

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