
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 $f(x) = \frac{a^x - a^{-x}}{a - 1}$ ($a > 0$ and $a \neq 1$).
(1) Determine the parity and monotonicity of $f(x)$;
(2) Given $p$: The inequality $af(x) \leq 2b(a+1)$ holds for any $x \in [-1, 1]$; $q$: The function $g(x) = \ln x - bx + 1$ ($b \in \mathbb{R}$) has a root. If $p$ or $q$ is true, and $p$ and $q$ are false, find the range of real numbers $b$.

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