
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 $f(x) = \ln(x + m) + n$, its graph has a tangent line at the point $(1, f(1))$ with the equation $y = x - 1$. The function $g(x) = ax^2 + bx$ $(a, b \in \mathbb{R}, a \neq 0)$ has an extreme value of $-2$ at $x = 2$.
(1) Find the explicit forms of the functions $f(x)$ and $g(x)$.
(2) If the function $y = f(x + 1) - g'(x)$ (where $g'(x)$ is the derivative of $g(x)$) is not monotonic in the interval $(t, t + \frac{1}{2})$, find the range of values for the real number $t$.
(3) Let $k \in \mathbb{Z}$, when $x > 1$, the inequality $k(x - 1) < xf(x) + 3g'(x) + 4$ always holds. Find the maximum value of $k$.

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