
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 the domain of $f(x)$ is $x∈R$ with a range of $[0,+∞)$.

(1) Find the expression for $f(x)$;
(2) Consider a function $g(x)=f(x)-2kx$, where $k∈R$.
   (a) If $g(x)$ is a monotonic function when $x∈[-2,2]$, find the range of values for the real number $k$;
   (b) If the minimum value of $g(x)$ on $x∈[-2,2]$ is $g(x)_{min}=-15$, find the value of $k$.

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