
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 $f(x)=ax^{2}+bx+c$ ($a > 0$, $b\in\mathbb{R}$, $c\in\mathbb{R}$).  
(I) If the minimum value of the function $f(x)$ is $f(-1)=0$, and $c=1$, and $F(x)= \begin{cases} f(x) & (x > 0) \\ -f(x) & (x < 0) \end{cases}$, find the value of $F(2)+F(-2)$;  
(II) If $a=1$, $c=0$, and $|f(x)|\leqslant 1$ holds true in the interval $(0,1]$, find the range of the real number $b$.

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