
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 $f(x)=x-a\ln x$, $g(x)=-\frac{1+a}{x}$ $(a > 0)$.

$(1)$ If $a=1$, find the extremum of $f(x)$;

$(2)$ If there exists $x_0 \in [1,e]$ such that $f(x_0) < g(x_0)$ holds, find the range of the real number $a$.

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