
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 $f(x)=ax^{2}+bx+1$ ($a > 0$), if $f(-1)=0$, and for any real number $x$, $f(x)\geqslant 0$ holds, let $g(x)=f(x)-kx$
$(1)$ When $x\in[-2,2]$, $g(x)$ is a monotonic function, find the range of the real number $k$;
$(2)$ When $x\in[1,2]$, $g(x) < 0$ always holds, find the range of the real number $k$.

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