
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 that the complex number $z\_1$ satisfies $(z\_1 - 2)(1 + i) = 1 - i$ ($i$ is the imaginary unit), the imaginary part of the complex number $z\_2$ is $2$, and $z\_1 \cdot z\_2$ is a real number. Find $z\_2$.

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