
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 $f(x)=ax\ln x+bx$ $(a\neq 0)$ has a tangent line at $(1,f(1))$ that is parallel to the $x$-axis,  
$(1)$ Discuss the monotonicity of $f(x)$ on $(0,+\infty)$;  
$(2)$ If there exists $a\in(e,+\infty)$, such that for any $x_{1},x_{2}\in\left[ \frac{1}{3}e,3e\right]$ the inequality $|f(x_{1})-f(x_{2})| < (m+e\ln 3)a+3e$ holds, find the range of the real number $m$ $(e=2.71828…)$.

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