
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 $f(x) = ax^2 + bx + c$ $(a > 0, b \in R, c \in R)$.

(1) If $a = c = 1$, $f(-1) = 0$, and $F(x) = \begin{cases} f(x), & \text{if } x > 0 \\ -f(x), & \text{if } x < 0 \end{cases}$, find the value of $F(2) + F(-2)$.

(2) If $a = 1$, $c = 0$, and $-1 \leqslant |f(x)| \leqslant 1$ always holds in the interval $(0,1]$, determine the range of values for $b$.

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