
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
Find the minimum value $g(a)$ of the function $f(x) = 1 - 2a - 2a\cos x - 2\sin^2 x$, where $a \in \mathbb{R}$.
(1) Find $g(a)$ when $a = 1$;
(2) Find $g(a)$;
(3) If $g(a) = \frac{1}{2}$, find $a$ and the maximum value of $f(x)$ in this case.

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