
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
Given a complex number $z_1$ satisfies $(z_1 - 2)i = 1 + i$,  
(1) Find $z_1$;  
(2) If the imaginary part of another complex number $z_2$ is 2, and $z_1 \cdot z_2$ is a real number, find the complex number $z_2$.

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