
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 $f(x) = ax^2 - |x| + 3a - 1$, where $a$ is a real constant,
(1) When $a = 0$, find the solution set of the inequality $f(2^x) + 2 \geq 0$;
(2) When $a < 0$, find the maximum value of the function $f(x)$;
(3) If $a > 0$, and the minimum value of $f(x)$ on the interval [1, 2] is denoted as $g(a)$, find the expression for $g(a)$.

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