
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 that for any real number $x$, the quadratic function $f(x) = x^2 - 4ax + 2a + 12$ ($a \in \mathbb{R}$) is always non-negative.
(1) Find the range of values for $a$;
(2) Find the range of values for the function $g(a) = (a + 1)(|a - 1| + 2)$.

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