
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 \( R \) in the complex plane is defined by:
\[ R = \{x + iy : -2 \leq x \leq 2, -2 \leq y \leq 2\}. \]
A complex number \( z = x + iy \) is chosen uniformly at random from \( R \). What is the probability that \( \left(\frac12 + \frac12i\right)z \) is also in \( R \)?

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