
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 the complex number \( z \) be such that the real and imaginary parts of both \( \frac{z}{10} \) and \( \frac{10}{\bar{z}} \) are positive and less than 1. Find the area of the region \( c \) that corresponds to these points in the complex plane.

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