
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 $z$ such that $z-1$ is purely imaginary and $\left(1-2i\right)\cdot z$ is real, where $i$ is the imaginary unit.<br/>$(1)$ Find the complex number $z$;<br/>$(2)$ If $x•z+y•\overline{z}=z•\overline{z}$, find the values of real numbers $x$ and $y$.

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