
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 $f(x)=\frac{1}{3}ax^3+\frac{1}{2}bx^2+cx (a > 0, b \in \mathbb{R}, c \in \mathbb{R})$, where $g(x)$ is the derivative of $f(x)$.
(1) If the minimum value of function $g(x)$ is $g(-1)=0$, and $c=1$, $h(x) = \begin{cases} g(x-1), & x\geqslant 1 \\ -g(x-1), & x < 1 \end{cases}$, find the value of $h(2)+h(-2)$;
(2) If $a=1$, $c=0$, and $|g(x)|\leqslant 1$ is always true in the interval $(0, 2]$, find the range of $b$.

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