
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

Let the function \( f(x) = \cos x \cos (x - \theta) - \frac{1}{2} \cos \theta \), where \( x \in \mathbb{R} \) and \( 0 < \theta < \pi \). It is known that \( f(x) \) attains its maximum value at \( x = \frac{\pi}{3} \).

(1) Find the value of \( \theta \).

(2) Let \( g(x) = 2 f\left(\frac{3}{2} x\right) \). Determine the minimum value of the function \( g(x) \) on the interval \(\left[0, \frac{\pi}{3}\right] \).

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