
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 with a range of $[-1, +\infty)$ satisfies $f(-1+x) = f(-1-x)$, and the two real roots of the equation $f(x) = 0$, $x_1$ and $x_2$, satisfy $|x_1 - x_2| = 2$.  
(1) Find the expression for $f(x)$.  
(2) The function $g(x) = f(x) - kx$ has its maximum value $f(2)$ and minimum value $f(-1)$ in the interval $[-1, 2]$. Find the range of the real number $k$.

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