
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 complex numbers $z_{1}=1+i$ and $z_{2}=x+yi$, where $x$ and $y$ are non-zero real numbers.  
$(1)$ If $z_{1}\cdot z_{2}$ is a real number, find the value of $\frac{x}{y}$.  
$(2)$ If $z_{2}=\overline{z_{1}}$, and the complex number $z=\left(\frac{z_{1}}{z_{2}}\right)^{2022}+(m^2-m-1)-(m+1)i$ is purely imaginary, find the value of the real number $m$.

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