
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 $f(x) = ax^2 + bx + 1$ that satisfies $f(-1) = 0$, and when $x \in \mathbb{R}$, the range of $f(x)$ is $[0, +\infty)$.  
(1) Find the expression for $f(x)$.  
(2) Let $g(x) = f(x) - 2kx$, where $k \in \mathbb{R}$.  
   (i) If $g(x)$ is monotonic on $x \in [-2, 2]$, find the range of the real number $k$.  
   (ii) If the minimum value of $g(x)$ on $x \in [-2, 2]$ is $g(x)_{\text{min}} = -15$, find the value of $k$.

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