
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
Let $y=f(x)$ be a function defined on the interval $(1, +\infty)$, with its derivative denoted as $y=f'(x)$. If there exists a real number $a$ and a function $y=h(x)$, where $h(x) > 0$ for any $x \in (1, +\infty)$, such that $f'(x) = h(x)(x^2 - ax + 1)$, then the function $y=f(x)$ is said to have property $P(a)$.
$(1)$ Let the function $f(x) = \ln x + \frac{b+2}{x+1}$, where $b$ is a real number.
($i$) Determine whether the function $y=f(x)$ has property $P(b)$, and explain the reason.
($ii$) Find the monotonic intervals of the function $y=f(x)$.
$(2)$ Given that the function $g(x)$ has property $P(2)$. For given $x_1, x_2 \in (1, +\infty)$ with $x_1 < x_2$, let $m$ be a real number, $\alpha = mx_1 + (1-m)x_2$, $\beta = (1-m)x_1 + mx_2$, and $\alpha > 1$, $\beta > 1$. If $|g(\alpha) - g(\beta)| < |g(x_1) - g(x_2)|$, find the range of values for $m$.

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