
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
Let the function be $f(x)=xe^{x}$.

$(1)$ Find the intervals of monotonicity and the extremum of $f(x)$;

$(2)$ Does there exist a real number $a$ such that for any $x_{1}, x_{2} \in (a, +\infty)$, when $x_{1} < x_{2}$, the inequality $\dfrac{f(x_{2})-f(a)}{x_{2}-a} > \dfrac{f(x_{1})-f(a)}{x_{1}-a}$ always holds? If it exists, find the range of $a$; if not, explain why.

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