
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 $f(x)=ax-\frac{1}{x}-(a+1)\ln x,\;\;a∈{R}$.<br/>$(1)$ When $a=0$, discuss the number of zeros of the function $f(x)$;<br/>$(2)$ If $a\geqslant 1$ and $f(x) \gt 1$ holds for all $x$ in the interval $[\frac{1}{e},e]$, find the range of values for $a$.

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