
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 $f(x)=x+ \dfrac{a}{x}+\ln x$, where $a\in \mathbb{R}$.

(Ⅰ) If $f(x)$ has an extremum at $x=1$, find the value of $a$;

(Ⅱ) If $f(x)$ is monotonically increasing in the interval $(1,2)$, find the range of $a$;

(Ⅲ) Discuss the number of zeros of the function $g(x)=f'(x)-x$.

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