
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 region \( S \) in the complex plane is defined by
\[
S = \{x + iy : -2 \leq x \leq 2, -2 \leq y \leq 2\}
\]
A complex number \( z = x + iy \) is chosen uniformly at random from \( S \). What is the probability that \( \left(\frac{1}{2} + \frac{i}{2}\right)z \) also lies in \( S \)?

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