
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 $ax^{2}+\left(a-1\right)x-1 \gt 0\left(a\in R\right)$.  
$(1)$ If the inequality holds when $x=-a$, find the range of real numbers for $a$;  
$(2)$ Solve this inequality in terms of $x$.

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