
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 $ax^{2}-x-b \gt 0\left(a,b\in R\right)$ with respect to $x$ is $\{x\left|\right.x \gt 2$ or $x \lt -1\}$.  
$(1)$ Find the values of $a$ and $b$;  
$(2)$ If $c\in R$, solve the inequality $ax^{2}-\left(ac+b-1\right)x+\left(b-1\right)c \lt 0$.

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