
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 $f(x) = \frac{\sqrt{5}}{a}x + \frac{\sqrt{5}(a-1)}{x}$, ($x \neq 0$) ($a \neq 0$).

(1) Discuss the monotonic increasing intervals of the function for different values of the real number $a$;  
(2) Given that when $a > 0$, the function is monotonically decreasing on the interval $(0, \sqrt{6})$ and monotonically increasing on the interval $(\sqrt{6}, +\infty)$, find the value of $a$ and write the analytic expression of the function;  
(3) If the function $f(x)$ has an inverse function in the interval $[-\frac{\sqrt{6}}{6}, 0) \cup (0, \frac{\sqrt{6}}{6}]$, find the range of values for the real number $a$.

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