
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 solution set of the inequality $x^{2}-\left(a+1\right)x+a \lt 0$ with respect to $x$ is $\{x\left|\right.b \lt x \lt 2\}$.  
$(1)$ Find the values of $a$ and $b$;  
$(2)$ When $x \gt 0$, $y \gt 0$, and satisfy $\frac{a}{x}+\frac{b}{y}=1$, if $x+2y≥\sqrt{k}-\frac{9}{\sqrt{k}}$ always holds, find the range of values for $k$.

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