
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 $f(x)=\log_{a}(\frac{1}{x}+a)$, where $a$ is a real number with $a > 0$ and $a \neq 1$.
$(1)$ If the function $g(x)=f(x)+\log_{a}{x^2}$ has a zero in the interval $(\frac{1}{2},\frac{3}{4})$, find the range of values for $a$.
$(2)$ Find all positive integers $m$ such that there exists $a \in (0,1)$, for any $x \in [m,7]$, the inequality $f(\frac{1}{{|ax-1|}}) > f(\frac{x}{{1-a}})$ holds.

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