
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 $f(x) = \frac {2}{\sqrt {kx^{2}+4kx+3}}$,
(1) If the domain of $f(x)$ is $\mathbb{R}$, find the range of the real number $k$;
(2) If the domain of $f(x)$ is $(-6, 2)$, find the value of the real number $k$;
(3) If the range of $f(x)$ is $(0, +\infty)$, find the range of the real number $k$.

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