
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 $f\left(x\right)=ax^{2}+x-a$, where $a\in \mathbb{R}$.  
$(1)$ If the inequality $f\left(x\right) \gt x^{2}+ax-1-a$ holds for all real numbers $x$, find the range of real numbers for $a$.  
$(2)$ If $a\in \mathbb{R}$, solve the inequality $f\left(x\right) \gt 1$.

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