
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 linear function $y_{1}=x-a+2$ intersects with an inverse proportion function ${y_2}=\frac{k}{x}(k≠0)$.  
$(1)$ Determine whether $y_{2}$ passes through the point $\left(k,1\right)$.  
$(2)$ If the graph of $y_{1}$ passes through the point $\left(k,1\right)$, and $2a+k=5$.  
① Find the expression of $y_{2}$.  
② Compare the values of $y_{1}$ and $y_{2}$ when $x \gt 0$.

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