
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 $f(x)=ax^2+bx+1$ ($a,b$ are real numbers, $a\neq 0$, $x\in \mathbb{R}$),

(1) If the graph of the function for $1\leqslant x_1 < x_2$ passes through the point $(-2,1)$, and the equation $f(x)=0$ has exactly one root, find the expression for $1\leqslant x_1 < x_2$;

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

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