
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 the function $f(x)$;

$(2)$ Determine whether there exist two different points on the graph of the function $f(x)$ such that the line passing through these two points is parallel to the $x$-axis;

$(3)$ Under what relationship between $a$ and $b$ will the function $f(x)$ always take positive values in the interval $(1,+\infty)$.

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