
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 that \(a\) is a positive integer and the equation \( \cos^{2} \pi(a-x) - 2 \cos \pi(a-x) + \cos \frac{3 \pi x}{2a} \cos \left(\frac{\pi x}{2a} + \frac{\pi}{3}\right) + 2 = 0 \) in terms of \(x\) has real solutions, find the smallest value of \(a\).

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