
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 $x$, $(ax-1)(x+1) > 0$.  
$(1)$ If the solution set of this inequality is $\left\{x|-1 < x < - \dfrac{1}{2}\right\}$, find the value of the real number $a$;  
$(2)$ If $a \leqslant 0$, solve this inequality about $x$.

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