
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 function $f(x)$ satisfies $f(\log_a x) = \frac{a}{a^2 - 1}(x - x^{-1})$, where $a > 0$ and $a \neq 1$,
1. Determine the evenness and monotonicity of the function $f(x)$;
2. For the function $f(x)$, when $x \in (-1, 1)$, $f(1-m) + f(1-m^2) < 0$, find the range of real numbers $m$;
3. When $x \in (-\infty, 2)$, the value of $f(x) - 4$ is always negative, find the range of $a$.

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