
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 $f(x)=ax-\ln x$ ($a\in \mathbb{R}$).

(1) When $a=1$, find the minimum value of $f(x)$;

(2) Given that $e$ is the base of the natural logarithm, and there exists $x\in\left[\frac{1}{e},e\right]$ such that $f(x)=1$ holds, find the range of values for $a$;

(3) If for any $x\in[1,+\infty)$, $f(x)\geqslant f\left(\frac{1}{x}\right)$ holds, find the range of values for $a$.

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