
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 $f(x) = ax - \ln(x)$, where $a \in \mathbb{R}$,
(1) Find the minimum value of $f(x)$ when $a=1$;
(2) Knowing that $e$ is the base of natural logarithms, and there exists $x \in \left[\frac{1}{e}, e\right]$ such that $f(x) = 1$, find the range of $a$;
(3) If for every $x \in [1, +\infty)$, we have $f(x) \geq f\left(\frac{1}{x}\right)$, find the range of $a$.

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