
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 $f(x)=\lg \frac{2x}{ax+b}$, $f(1)=0$, when $x > 0$, it always holds that $f(x)-f\left( \frac{1}{x}\right)=\lg x$.  
$(1)$ Find the expression and domain of $f(x)$.  
$(2)$ If the equation $f(x)=\lg t$ has a solution, find the range of the real number $t$.  
$(3)$ If the solution set of the equation $f(x)=\lg (8x+m)$ is the empty set, find the range of the real number $m$.

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