
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 $f(x)=ax-\frac{2a}{x}-3\ln x$.

(1) Find the value of the real number $a$ if the function $f(x)$ has an extreme value at $x=1$.

(2) Under the condition of (1), find the maximum and minimum values of $f(x)$ on $[1,e]$.

(3) Find the range of values of the real number $a$ if the function $f(x)$ is an increasing function on its domain.

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