
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 $f(x)=x\ln x+a(x^{3}-x)$.  
$(1)$ Discuss the monotonicity of $\frac{f(x)}{x}$;  
$(2)$ Given $g(x)=2x-e^{x-1}-1$, if $f(x)\geqslant g(x)$ always holds, find the value of $a$.

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