
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 $\left(\frac{1}{2} + \frac{1}{2}i\right)z$ is also in $R$?

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