
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
Let the function $f(x)=(x+1)\ln x-a(x-1)$ have a tangent line at $x=e$ that intersects the $y$-axis at the point $(0,2-e)$.  
(1) Find the value of $a$;  
(2) Can the function $f(x)$ have an extremum at $x=1$? If it can, find this extremum; if not, explain why.  
(3) When $1 < x < 2$, compare the size of $\dfrac{2}{x-1}$ and $\dfrac{1}{\ln x}- \dfrac{1}{\ln (2-x)}$.

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