
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 function $ f(x) = 1 - 2a - 2a\cos x - 2\sin^2 x $'s minimum value $ g(a) $ where $ a \in \mathbb{R} $.
1. Calculate $ g(a) $.
2. If $ g(a) = \frac{1}{2} $, find $ a $ and the maximum value of $ f(x) $ at this condition.

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