
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 $f(x) = \lg(a^x - b^x)$ where $a > 1 > b > 0$.

(1) Find the domain of $y = f(x)$.

(2) Determine whether there exist two distinct points on the graph of $y = f(x)$ such that the line passing through these two points is parallel to the x-axis.

(3) Under what conditions for $a$ and $b$ does $f(x)$ always take positive values on the interval $(1, +\infty)$.

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