
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
Let the solution set of the inequality $ax+b \gt 0$ be $A$, and the solution set of the inequality $\left(ax-b\right)\left(x-a\right)\geqslant 0$ be $B$.  
$(1)$ Given $a\in R$ and $b=-2a$, find $A$;  
$(2)$ If $A=\left(-\infty ,1\right)$, find $B$.

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