
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 $f(x)=x\ln x$ and $g(x)=x^{3}+ax^{2}-x+2$.

$(1)$ Find the intervals of monotonicity for $f(x)$.

$(2)$ If for any $x\in(0,+\infty)$, $2f(x)\leqslant g′(x)+2$ always holds, find the range of the real number $a$.

$(3)$ Let $h(x)=f(x)-a(x-1)$, where $a\in\mathbb{R}$, find the minimum value of the function $h(x)$ on the interval $[1,e]$.

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