
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 $f(x)=ax^{2}+bx+1(a\neq 0,\,b\in\mathbb{R})$, if $f(-1)=0$, and for any real number $x(x\in\mathbb{R})$, the inequality $f(x)\geqslant 0$ always holds.

1. Find the values of real numbers $a$ and $b$.
2. When $x\in[-2,2]$, $g(x)=f(x)-kx$ is a monotonic function, find the range of real number $k$.

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