
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 $x^{2}-(a+2)x+b\leqslant 0$ is $\{x|1\leqslant x\leqslant 2\}$.  
$(1)$ Find the values of real numbers $a$ and $b$;  
$(2)$ Solve the inequality with respect to $x$: $(x-c)(ax-b) \gt 0(c$ is a constant, and $c\neq 2)$.

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