
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 $\vec{a} = \{\cos \alpha, \sin \alpha\}$ and $\vec{b} = \{\cos \beta, \sin \beta\}$ such that $|k \vec{a} + \vec{b}| = \sqrt{3} \mid \vec{a} - k \vec{b} \mid (k > 0)$:
1. Express $\vec{a} \cdot \vec{b}$ in terms of $\alpha$ and $\beta$.
2. Express $\vec{a} \cdot \vec{b}$ in terms of $k$.
3. Find the minimum value of $\vec{a} \cdot \vec{b}$ and the corresponding angle $\theta$ between $\vec{a}$ and $\vec{b}$ $(0 \leqslant \theta \leqslant \pi)$.

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