
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 $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 1$, and $|k\overrightarrow{a} + \overrightarrow{b}| = \sqrt{3}|\overrightarrow{a} - k\overrightarrow{b}|$ where $k > 0$,  
(1) Express $\overrightarrow{a} \cdot \overrightarrow{b}$ in terms of $k$ and find the maximum value of the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{b}$.  
(Note: If $a > 0$, $b > 0$, then $a + b \geq 2\sqrt{ab}$, with equality if and only if $a = b$.)  
(2) If $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of the real number $k$.

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