
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 $f(x)=4ax-\frac{a}{x}-2\ln x$.

(1) When $a=1$, find the equation of the tangent line to the curve $f(x)$ at the point $(1,f(1))$;

(2) If the function $f(x)$ is an increasing function in its domain, find the range of values for the real number $a$;

(3) Let $g(x)=\frac{6e}{x}$. If there exists at least one point $x\_0$ in the interval $[1,e]$ such that $f(x\_0) > g(x\_0)$, find the range of values for the real number $a$.

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