
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 $\boldsymbol{a} = (\cos \alpha, \sin \alpha)$ and $\boldsymbol{b} = (\cos \beta, \sin \beta)$, the relationship between $\boldsymbol{a}$ and $\boldsymbol{b}$ is given by $|k \boldsymbol{a} + \boldsymbol{b}| - \sqrt{3}|\boldsymbol{a} - k \boldsymbol{b}|$, where $k > 0$. Find the minimum value of $\boldsymbol{a} \cdot \boldsymbol{b}$.

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