
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 function $f(x)=ax^2+bx+1$ (where $a, b$ are real numbers), $x\in\mathbb{R}$, and $F(x) = 
\begin{cases} 
f(x) & \text{for } x > 0 \\
-f(x) & \text{for } x < 0
\end{cases}$.  
(1) If $f(-1) = 0$ and the range of function $f(x)$ is $[0, +\infty)$, find the expression for $F(x)$.  
(2) Under the condition of (1), given $x\in[-2, 2]$, if $g(x) = f(x) - kx$ is a monotonic function, find the range of values for real number $k$.

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