
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 $f(x)={\log_3}({\frac{2}{x}-a})$.  
$(1)$ When $a=1$, find the solution to the inequality $f(x) \lt 0$ in terms of $x$.  
$(2)$ Determine whether the function $g(x)=f(x)-{\log_3}(ax+a-1)$ could have two zeros, and explain the reason.  
$(3)$ Let $a \lt 0$. If for any $t∈[{\frac{1}{4},1}]$, the difference between the maximum and minimum values of the function $f(x)$ on the interval $\left[t,t+1\right]$ does not exceed $1$, find the range of real numbers for $a$.

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