
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 \( f(x) \) satisfying \( f\left(\log_{a} x\right)=\frac{a}{a^{2}-1}\left(x-x^{-1}\right) \), where \( a > 0 \) and \( a \neq 1 \):
1. Determine the even or odd nature and the monotonicity of the function \( f(x) \).
2. When \( x \in (-1, 1) \), find the range of the real number \( m \) such that \( f(1 - m) + f\left(1 - m^{2}\right) < 0 \).
3. When \( x \in (-\infty, 2) \), find the range of \( a \) such that the value of \( f(x) - 4 \) is always negative.

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