
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 $i$ be the imaginary unit, and let $a+bi= \frac{(1-i)^{2}+2(5+i)}{3+i}$ where $a,b \in \mathbb{R}$.
(I) Find the values of $a$ and $b$;
(II) Let $z=-1+yi$ where $y \in \mathbb{R}$, and suppose that the point corresponding to the complex number $(a+bi)z$ lies on the bisector of the first and third quadrants in the complex plane. Find $z$.

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