
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 $f(x) = ax^2 + bx + c$ ($a > 0, b \in \mathbb{R}, c \in \mathbb{R}$),
1. If the minimum value of the function $f(x)$ is $f(-1) = 0$, and $c = 1$, 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 $|f(x)| \leq 1$ holds for the interval $(0, 1]$, determine the range of values for $b$.

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