
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 \( \left(z_{1}-2\right)(1+\mathrm{i})=1-\mathrm{i} \) (where \( \mathrm{i} \) is the imaginary unit), and that the imaginary part of the complex number \( z_{2} \) is 2, find the condition for \( z_{2} \) so that \( z_{1} \cdot z_{2} \) is a real number.

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