
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 functions $f(x)=x-(a+1)\ln x$ and $g(x)= \frac{a}{x}-3$ $(a\in\mathbb{R})$.

$(1)$ Let $h(x)=f(x)-g(x)$, discuss the monotonicity of the function $h(x)$;  
$(2)$ If for any $x\in[1,e]$, $f(x)\geqslant g(x)$ always holds, find the range of the real number $a$.

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