
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 $f(x)=ax^{2}+bx+1$ where $a,b$ are real numbers and $x\in R$. Let $F(x) = \begin{cases} f(x), & x > 0 \\ -f(x), & x < 0 \end{cases}$.

(1) If $f(-1)=0$ and the range of $f(x)$ is $[0,+\infty)$, find the expression for $F(x)$.

(2) Under the conditions in (1), when $x\in[-2,2]$, $g(x)=f(x)-kx$ is a monotonic function. Find the range of values for the real number $k$.

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