
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 $x, y \in \mathbb{R}$, and let the complex number $z$ be defined as $z = x + yi$. Suppose $z$ satisfies the equation $|z|^2 + (z + \bar{z})i = \frac{3-i}{2+i}$. Find the values of $x$ and $y$.

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