
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 the function $f(x) = 3 - 2a - 2a\cos x - 2\sin^2x$,

(1) Find the expression for the minimum value $g(a)$ of the function $f(x)$;

(2) Find the value(s) of $a$ that make $g(a) = \frac{5}{2}$, and find the range of $k$ that makes $f(x) - k \leq 0$ hold true for all $x \in \mathbb{R}$ when $a$ takes these values.

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