
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 function $f\left(x\right)=\sqrt{|x+a|-a}-x$.  
$(1)$ If $a=1$, find the domain of the function.  
$(2)$ If $a\neq 0$ and $f\left(ax\right)=a$ has $2$ distinct real roots, find the range of $a$.  
$(3)$ Does there exist a real number $a$ such that the function $f\left(x\right)$ is monotonic within its domain? If yes, find the range of $a$.

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