
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 a quadratic function $f(x) = ax^2 + bx + 1$ ($a>0$), and $F(x)$ is defined as follows: if $f(-1) = 0$, and for any real number $x$, $f(x) \geq 0$ holds.  
(1) Find the expression for $F(x)$.  
(2) When $x \in [-2,2]$, $g(x) = f(x) - kx$ is a monotonic function, find the range of values for $k$.

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