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

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