
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 power function $f(x) = x^{-k^2 + k + 2}$, where $k \in \mathbb{Z}$, and it satisfies $f(2) < f(3)$.
(1) Find the value of the real number $k$ and the corresponding function expression $f(x)$;
(2) For the function $f(x)$ in (1), determine whether there exists a positive number $q$ such that the function $g(x) = 1 - qf(x) + (2q - 1)x$ has a range of $[-4, \frac{17}{8}]$ on the interval $[-1, 2]$. If it exists, find this $q$.

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