
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 $a > 1$, and $f(\log_a x) = \frac{a}{a^2 - 1}(x - \frac{1}{x})$.

(1) Find the explicit formula for $f(x)$;  
(2) Determine the parity and monotonicity of $f(x)$ (conclusions only, no proof required);  
(3) For $f(x)$, when $x \in (-1, 1)$, if $f(1-m) + f(1-m^2) < 0$, find the range of $m$.

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