
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 a function $f(x) = x + \frac{a}{x} - (a-1)\ln x - 2$, where $a \in \mathbb{R}$.  
(1) If $f(x)$ has a unique critical point and the extremum is 0, find the value of $a$.  
(2) Discuss the number of zeros of $f(x)$ on the interval $[1,e]$.

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