
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 vectors $\overrightarrow{a} = (\cos\alpha, \sin\alpha)$, $\overrightarrow{b} = (\cos\beta, \sin\beta)$, and $\overrightarrow{a}$, $\overrightarrow{b}$ satisfy the relation $|k\overrightarrow{a} + \overrightarrow{b}| = \sqrt{3}|\overrightarrow{a} - k\overrightarrow{b}|$ (where $k$ is a positive number).  
(1) Find the dot product of $\overrightarrow{a}$ and $\overrightarrow{b}$ expressed as an analytical formula $f(k)$ in terms of $k$.  
(2) Can $\overrightarrow{a}$ be perpendicular to $\overrightarrow{b}$? Can $\overrightarrow{a}$ be parallel to $\overrightarrow{b}$? If not, explain why; if yes, find the corresponding value of $k$.

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