
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
The function $f(x)=x^{3}+ax^{2}+x+1$, $x\in R$. It is known that $x=x_{1}$ and $x=x_{2}$ are the local maximum and local minimum points of the function $f(x)$, respectively.  
$(1)$ Find the range of real number $a$;  
$(2)$ Find the range of $f(x_{1})-f(x_{2})$.

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