
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 the inequality about $x$: $ax^{2}-b\geqslant 2x-ax$ $(a,b\in R)$.  
$(1)$ If the solution set of the inequality is $\{x|-2\leqslant x\leqslant -1\}$, find the values of $a$ and $b$.  
$(2)$ If $a \lt 0$, solve the inequality $(ax-2)(x+1)\geqslant 0$.

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