
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
The minimum value of the function $f(x)=1-2a-2a\cos x-2\sin ^{2}x$ is $g(a)$ $(a\in R)$.

$(1)$ Find $g(a)$;

$(2)$ If $g(a)= \dfrac{1}{2}$, find $a$ and the maximum value of $f(x)$ at this time.

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