
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 any \(\alpha, \beta \in \left(0, \frac{2\pi}{3}\right)\), the following inequality holds:
\[ 
4 \cos^2 \alpha + 2 \cos \alpha \cdot \cos \beta + 4 \cos^2 \beta - 3 \cos \alpha - 3 \cos \beta - k < 0.
\]

Find the minimum value of \(k\).

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